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https://ntrs.nasa.gov/search.jsp?R=19720005302 2020-05-18T17:30:26+00:00Z
Aerotherm Project 6099
Aerotherm Final Report 71-38
INVESTIGATION OF THERMAL PROTECTION SYSTEMS EFFECTS
ON VISCID AND INVISCID FLOW FIELDS FOR MANNED EMTRY SYSTEMS
by Eugene P. Bartlett Howard L. horse
Henry Tong
prepared for
National Aeronautics and Space Administration
September 22, 1971
Extension to
Contract NAS9-9494
Technical Management NASA Manned Spacecraft Center
Bouston, T e x a s Structures and Mechanics Division
Donald M. Curry
FOREWORD
This report was prepared by the Aerot1,iim Division of the Acurex Corpora- . tion under an extension of NASA/Manned Spacecraft Center Contract NAS9-9494.
The period covered by this extension was from 9 March 1971 to 9 September 1971.
The sponsor of the program was the Thermal Protection Section, Structures
and Mechanics Division, Manned Spacecraft Center, National Aeronautics and Space
Administration, Houston, Texas. Mr. Donald M. Curry was the NASA/MSC technical
monitor.
The Aerotherm program manager and p~incipal investigator was Eugene P.
Bartlett and this report was prepared with contributions from Howard L. Morse
and Henry Tong.
ABSTRACT
Procedures and metlods for predicting aerothermodynamic heating to
delta orbiter shuttle vehicles has been reviewed. A number of approximate
methods were found to he adequate for large scale parameter studies, but are
considered inadequate for final design calculations. It is recommended that final
design calculations be based on a computer code which accounts for non-
equilibrium chemistry, streamline spreading, entropy swallowing, and turbulence
It is further recommended that this code be developed with the intent that it
can be directly coupled with an exact inviscid flow field calculation when the
latter becomes available.
The recommended procedure for parameter studies is to calculate local
pressures based on tangent cone or wedge approximations and heat transfer
following Eckert's reference enthalpy method. This procedure is relatively
simple and was found to agree favorably with wind tunnel data for a shuttle
configuration at angles-of-attack which are of potential interest.
A nonsimilar, equilibrium chemistry computer code (BLIMP) was used to
evaluate the effects of entropy swzllowing, turbulence, and various three
dimensional approximations. These solutions were compared with available
wind tunnel data. It was found from this study that, for wind tunnel conditions,
the effect of entropy swallo.ring and three dimensionality are small for laminar
boundary layers but entropy swallowing causes a significant increase in turbulent
heat transfer. However, it is noted that even small effects (say, 10-20%) may
be important for the shuttle reusability concept.
iii
ri' :: . . I . .. - .- _
TABLE OF CONTENTS
Section Page
1 INTRODUCTION
2 CHARACTER OF FLOW FIELD DURING HEATING 2.1 Stagnation Region 2.2 Downstream Regions
3 APPROXIMATE FIEAT TRANSFER PR ICTION METHODS 3.1 Stagnation and Leading Eage Regions 3.2 Downstream Regions
3.2.1 Inviscid Flow 3.2.2 Boundary Layer
4 CURFtENT METHODOLOGY 4.1 Windward Surface 4.2 Leeward Surface 4.3 Shortcomings of Current Methodology 4.4 Recommended Current State-of-the-Art
5 CANDIDATE ADVANCED BOUNDARY LAYER TECHNIQUES 35
6 BLIMP PREDICTION CAPABILITY 6.1 Pressure Data for BL.MP Input 6.2 Model Geometry 6.3 Shock Wave Geometry for Entropy Layer Predictions 6.4 Approximation of Three Dimensional Flow Effects 6.5 BLIMP Prediction Matrix 6.6 Results and Discussion of BLIMP Studies
7 RECQMMENDATIONS FOR FURTHER DEVELOPMENT OF HEATING PREDICTION CAPABILITY FOR SHUTTLE VEHICLES 62
REFERENCES 63
APPENDIX 1 - LOCAL SURFACE INCIDENCE RELATIVE TO FREE STREAM DIRECTION
APPENDIX 2 - INPUT DATA FOR BLIMP TEST CASES AP?ENDIX 3 - SPREADING FACTOR FOR NONEQUAL BODY CURVATURE
Number
1 Page
Recommended Methods for Approximate iieat Transfer Predictions to Delta Shutt-le Vehicles
Test Conditions for the Predicted Cases
Matrix of BLIMP Predictions
LIST OF FIGURES
Number
1
2
Page
16 Typical Heating Zones on Delta Vehicle
Comparison of Predicted and Measured Pressures on Windward Generator of NAR Yodel 134B
Comparison of Tangent Wedge and Newtonian-Tangent Wedge Pressure on Wing of NAR Model 134B at 20% Semi-span, a = 30° Laminar Flow
Comparison of Predicted and Measured Pressure on Wing of NAR Model 134B, a = 30° Laminar Flow
Body Reference Angles
Comparison of Predicted and Measured Heat Transfer on Wind- ward Generator of NAR Model 134B, a = lSO Laminar Flow
Comparison of Predicted and Measured Heat Transfer on Wind- ward Generator of NAR Elodel 134B, a = 30° Laminar Flow
Representative Profiles of NAR Model 134B
Windward Generator Heating for the NAR Delta Wing Shuttle Model 134B, a = 15O Laminar Flow
Windward Generator Heating for the NAR Delta Wing Shuttle Model 134B, a = 30° Laminar Flow
Windward Generator Heating for NAR Delta Wii~g Shuttle Model 134B, a = 30° Turbulent Flow
Windward Generator Heating for the NAR Delta Wing Shuttle Model 134B, a = 54.50 Laminar Flow
Shock Angle o.E Streamline in Vicinity of Boundary Layer Edge
Velocity and Streamline 3rigin Pzofiles, a = 15O Laminar Flow
Velocity and Streamline Origin Profiles, a = 30° Laminar Flow
Velocity and Streamline Origin Profiles, a = 30° Turbulent Flow
Comparison of BLIMP and Tangent Cone Approximations, a = 15O Laminar Flow
Comparison of BLIMP and Tangent Cone Approximation, a = 30° Laminar Flow
Comparison of BLIMP and Tangent Cone Approximation, a = 53.5' Laminar Flow
LIST OF SYMBOLS
angle between fin chord line cnd vehicle syrrunetry plane (see F.'q. 5 )
specific heat capacity
skin friction coefficient
heat transfer coefficient
spreading factor
form factor = displacement thickness/momentum thickness
enthalpy
thermal conductivity
Reynolds analogy factor, alsov< meridianal coordinate
pressure
heat transfer rate
heat transfer coefficient
recovery factor
normal distance from centerline to body surface for an axisymmetric body
nose radius
streamwise radius
transverse radius
streamwise coordinate
streamwise normal velocity
cross flow velocity
coordinates 0
Greek Symbols - vehicle reference line angle of attack
angle between wing chord line and refexence line
local surface inclination with respect to chord line
norminal sweep angle of fin (see Fig. 5 )
true angle of incidence on wing; also specific heat ratio
true angle of incidence on fin
semi-apex angle of wing (see Fig. 5)
body surface inclination
true sweep angle of wing leading edge (see Fig. 5 )
true sweep angle of fin leading edge (see F i g . 5 )
viscos. . y
density
local wing dihedral angle (see Fig. 5)
tilt angle of fin (see Fig. 5 )
Subscripts
dissociation
boundary layer edge
stagnation
reference
stagnation point or stagnation line
freestream
Dimensionless Parameters
Lewis number
Mach number
Nusselt number
Prandtl number
Stanton number
Reynolds number
Reynolds number based on momentum thickness
vii
SECTION 1
SNTRODUCTION
* ' I . v:;
. ..f
The design of an efficient thermal protection system for a shuttle orbi-
ter requires reliable heat transfer prediction methods. Under orbital entry
conditions, heating is principally a boundary layer phenomena but even exact
boundary layer analyses would be inadequate without sufficient methods for pre-
dicting pressure distributions and edge boundary conditions. These predictions
of aerodynamic heating and edge conditions be based on analytic methods,
experimental data or a hybrid combination of these two.
Analytical techniques, which are presently available. cannot yield pre-
cise predictions for heating distributions everywhere on ',he body although, with
suitable approximations, they can satisfactorily pr~dict heatinq rates to local
regions such as the nose, leading edge and centerline p~rtion of the orbiter.
The key here is "suitable approximations.'' The test of whether 01: not an approx-
imate is suitable depends on a comparison of predictions with eith c an exact solution or experimental data. An exact solution for shuttle geo!netries is not
available and wind tunnel data, though highly desirable, is singularly insuffi-
cient becauss of inadequate simultanenus duplication of pertinent parameters in
ground test facilities and the lack of complete model scaling laws to permit
confidence in extrapolating to flight conditions.
Thus the present capability must rely on approximate analytic or semi-
analytic methods fortified by ground facility experimental data. The ideal kind
of analysis would be one which is derived from a consideration of all phenomena
(including for example, homogeneous chemistry) associated with the flight condi-
tions and which can be generalized to wind tunnel conditions. Then if the analy-
sis adequately predicts the levels of wind tunnel heating, there will be a high
degree of confidence in the predicted heating to the flight vehicle.
A very comprehensive review on heat transfer prediction capabilities for
Apollo-class vehicles is presented in Reference 1. However, shuttle differs
from Apollo in many aspects, each of which increases the prediction requirements.
The present discussion will center around these difficulties and prediction
methodoiogy which are directly applicable to shuttle requirements. Two of the
most outstanding differences between shuttle and Apollo are that shuttle is a
true three-dimensional body and is large; a typical configuration would cover
ha l f of a f o o t b a l l f i e l d . An a d d i t i o n a l d i f f e r e n c e which i s not r e a d i l y appar-
e n t i s i n t h e des ign philosophy; each Apollo was flown only once s o t h a t it was,
a t l e a s t i n p r i n c i p l e , pos s ib l e t o design e a r l y veh ic l e s conservc t ive ly and ad- j u s t t h e thermal p ro t ec t ion weights on success ive veh ic l e s us ing f l i g h t da t a . Such i s no t t h e ca se with s h u t t l e . To be succes s fu l , t h e " f i r s t - o f f " s h u t t l e should be designed f o r upwards of 100 f l i g h t s s o t h a t u l t ra -conserva t ive design
procedures would render s h u t t l e imprac t ica l .
The choice of a thermal p r o t e c t i o n system has a s i g n i f i c a n t bear ing on t h e requi red p r e d i c t i o n accuracy. Ablat ive m a t e r i a l s , a s were used on Apollo, have a high degree of accommodation o r t o l e r a n c e and can surv ive through sus- t a ined exposure t o higher-than-expected hea t ing r a t e s s o long a s s u f f i c i e n t coa t ing tk lcknesses a r e provided. I n c o n t r a s t , t h e su r f ace temperature gf a
r a d i a t i o n cooled su r f ace , as i s proposed f o r s h u t t l e a p p l i c a t i o n s , i s dependent
on t h e hea t ing r a t e s . M e t a l l i c s i x f a c e s have a maximum s e r v i c e ten1peratu:e which when exceeded r e s u l t s i n r a p i d d e t e r i o r a t i o n of t h e m a t e r i a l and w i l l re-
duce t h e degree of veh ic l e r e u s a b i l i t y . Hence p r e d i c t i o n accuracy i s more c r i - t i c a l f o r s h u t t l e .
The s e l e c t i o n of m a t e r i a l s f o r t h e s h u t t l e thermal p r o t e c t i o n s l s tem de-
pends on the expected peak hea t ing r a t e s which w i l l be maintained lower than those experienced by Apollo. This w i l l be achieved by d e c e l e r a t i n g t h e veh ic l e
a t h igher a l t i t u d e s bu t , a t t h e lower d e n s i t i e s a s soc i a t ed wi th t h e s e a l t i t u d e s , non-equilibrium chemistry e f f e c t s a r e enhanced. These e f f e c t s may be b e n e f i c i a l i n . the s t agna t ion reg ion i f nonca t a ly t i c m a t e r i a l s a r e u t i l i z e d b u t a penal ty must be pa id i n increased dcwnstream hea t ing . Nonequilibrium chemistry a l s o in-
c r eases t h e concent ra t ion of d i s s o c i a t e d oxygen near t h e s u r f a c e which may dras- t i c a l l y i nc rease ox ida t ion r a t e s and hence redace t h e i i f e expectancy of t h e veh ic le .
S h u t t l e s i z e enhances t h e p r o b a b i l i t y of t u r b u l e n t flow i r . : ? bo~ndaxy
l aye r which i n t u r n i nc reases hea t ing r a t e s . A serioixi ciuestioa . s i , s e d re-
garding where on t h e veh ic l e and when dur ing t h e t r a j e c t o r y t r a n s i t i o n w i l l
occur. T r a n s i t i o n dur ing t h e peak .hea t ing phase of t h e ei.try may d i c t a t e hiqher temperature m a t e r i a l s on t h e s h u t t l e af terbody. Thus adequate t r a n s i t i c n c r i - ter ia must be s p e c i f i e d .
Apollo was launched i n tandam with t h e launch v e h i c l e and thermal ly pro-
t e c t e d from ascen t heat ing. S h u t t l e w i l l probably be launched i n a piggy-back
fash ion s o t h a t shock i n t e r f e r e n c e between t h e launch v e h i c l e and s h u t t l e may
l o c a l l y i nc rease t h e thermal p r o t e c t i o n requirements. Fu r the r , dur ing entcy maneuvers, var ious c o n t r o l s u r f a c e s w i l l i n t e r a c t wi th t h e normal flow f i e l d
p a t t e r n s t o cause i n t e r f e r e n c e hea t ing and flow separat ion-reat tachment problems.
SECTICN 2
CHARACTZR OF FLOW FIELD DURING HEATING
For purposes of convenience and analytic simplification, assumptions re-
garding the nature of a hypersonic flow field are often divided into the fluid flow and chemistry aspects. Typical assumptions which are made are shown below.
I Hypersonic Flow Field 1'
Viscous =7 Inviscid
Laminar Turbulent I* -- quiiibrium
Separation I 1 Nonequilfbrium T p z q Kinetics
Depending on the entry conditions and location on the vehicle, various combi1.a-
tions may be simultaneously important. For example, at the stagnation point,
a laminar boundary layer with n~nequilibrium chemistry may occur. The challenge
of shuttle is that somewhere in its trajectory tliere will be separated, inviscLd,
,?.nd .risoous flows; the Last will involve laminar and turbulent flows and non-
equilibrium chenical kinetics will be important. In short, all of the above
phenomena at one time or anotter will be important.
2.1 STAGNATION REGION
In the stagnatior ~egion of the nose or leading edge and at low altitudes
where the density is high, the chemical reaction rates will be rapid enough for
equilibrium assumptions to be valid. Further it can be shown that there are
negligibly small errors associated with the assumptioc that a well defined bound-
aq- layer _xists with zero gradients for edge conditions. Within the confines
of these assumptions, several solutions (analytic, correlation and experimental)
are available and have prove- accuracy.
At moderate altitudes, the equilibrium chemistry assumption is not valid
although the boundary layer assumptions still are. Thus the viscous and invis-
cid fields are still decoupled and can be analyzed separately. In the inviscid
shock layer, which is much thicker than the boundary layer, the predominant
chemical reaction is dissociation. Dissociation, being a two-body reaction, can
ac.rieve equilibrium even though recombination in the boundary layer is near fro-
zen. Thus the boundary layer edge conditions can be readily determined without
solving the inviscid field. Because of this simplification, analyses are avail-
able for the nonequilibrium stagnation boundary layer. A useful sirn;?lification
and special case of nonequilibrium is a frazen boundary layer. This assumption
has often been used for studies of surface kinetics.
At high altitudes, even dissociation rates are nat rapid enough to achieve
equilibrium, so if boundary layer assumptions are retained a more careful analy-
sis of th. .inviscid flow must be used to. determine the boundary layer edge con-
ditions. Actually, the boundary layer assumptions commence to breakdown as non-
equilil.rium chertistry begins to be important so that at high altitudes nonequi-
librium viscous shock layer assumptions are used for analysis. Solution methods
for these flows are available for simplified air models and recently for general
multicomponent models.
At still higher altitudes, the shock wave begins to grow and can no longer
be considered as thin. Eventually tbe shock wave merges with the shock layer to
form a viscous merged layer; General chemistry analyses of this flow regime are
not available but, fortunately, it does not occur during the shuttle heating
cycle.
2.2 DOWNSTREAM REGIONS
Regardless of t ' *ther the stagnation flow is of the boundary layer type
or viscous shock layer type, f3r enough 4ownst.ream the windward side of shuttle
will agpr9ach a viscous and an inviscid layer. Because of the chemical reactions
and e f f e c t s of shock cu rva tu re , t h e governing d i f f e r e n t i a l equa t ions , wi th
boundary l a y e r assumptions, are nonsimilar b u t can s t i l l be analyzed provided
t h e edge boundary cond i t i ons are known.
However, a t high a l t i t u d e s and j u s t downstream o f t h e s t agna t ion reg ion ,
t h e boundary l a y e r assumptions may n o t be v a l i d . For t h e s e cond i t i ons , t h e com-
p l e t e shock l a y e r may be v i scous , chemically r e a c t i n g an3 nonsimilar , a condi-
t i o n f o r which t h e r e a r e apparen t ly co publ ished exac t solutio1.s.
When t h e boundary l a y e r assumptions a r e v a l i d , s o l u t i o n s are a v a i l a b l e
f o r gene ra l a i r , chemically r e a c t i n g f lows around simple body shapes. These
techniques show promise, wi th t h e a p p l i c a t i o n of s t r eaml ine divergence methods,
o f being s u i t a b l e f o r s h u t t l e , Nonetheless, even a p e r f e c t boundary l a y e r
a n a l y s i s would be inadequate i f app rop r i a t e techniques are n o t a v a i l a b l e f o r
spec i fy ing t h e edge boundary condi t ions . P re s su re d i s t r i b u t i o n s do n o t seem
t o be very s e n s i t i v e t o chemistry s o t h e y are o f t e n determined a p r i o r i , by approximation techniques , Subsequently, t h e o t h e r edge cond i t i ons a r e approxi-
mated. The ideal approach would be to s o l v e the d i r e c t problem f o r t h e i nv i s -
c i d f i e l d , bu t p r e s e n t technology* permi t s t h i s only f o r simple body shapes and
even thcn c a l c u l a t i o n t i m e s are long and n c t p r a c t i c a l f o r extended t r a j e c t o r y
s t u d i e s .
As the flow progresses downstream, boundary l a y e r growth may lead t o
t r a n s i t i o n i n t o t u r b u l e n t flow. Although c o r r e l a t i o n and analogue methods are .
a v a i l a b l e f o r approximate p r e d i c t i c n s of t u r b u l e n t hea t ing , t h e p r e d i c t i o n of
when t r a n s i t i o n w i l l occur i s i n a s t a t e of f l u x . The es tab l i shment of a t r a n -
s i t i o n cri teria is f u r t h e r complicated by 4-ts dependence on t h e s t a t e of t h e
gas a t t h e edge of t h e boundary l a y e r and ocher boundcry l a y e r parameters.
If t h e windward s i d e of s h u t t l e poses sone d i f f i c u l t p r ed i c t i on prob-
l e m s , accura te p r e d i c t i o n of t h e leeward side i s v i x t u a l l y i ~ p o s s i b l e . Only
a very l imi t ed number of s t u d i e s have been conducted on leeward s h u t t l e heat-
ing. A t very s m a l l angles-of-at tack, when t h e flow remains a t tached , methods similar t o t hose Used f o r t h e windward s i d e a r e appl icab le , al though it should
be noted t h a t he re t h e s t reaml ines are cor~verging r a t h e r than diverging as on
t h e windward s ide . However, a t high m g l s a of a t t a c k , f low sepa ra t ion occurs and
t h e viscous- inviscid i n t e r a c t i o n s a r e n o t y e t amenable t o ana lys i s . Extensive
experimental d a t a and conserva t ive e x t r a p o l a t i o n s appear t o be t h e only recourse.
* I n t h i s con tex t , "presen t technologyM means e x i s t i n g , o p e r a t i o n a l and reliable computer code 3.
SECTION 3
APPROXIMATE HEAT TRANSFER PREDICTION METHODS
3.1 STAGNATION AND LEADING EDGE REGI9NS
The s ta te -of - the-a r t f o r p r e d i c t i n g s t agna t ion region h e a t t r a n s f e r
rates t o planae or axisymmetric bodies is w e l l I n hand. Adequate p r e d i c t i o n s
can be ruade using a v a i l a b l e solutions2-" f o r both non-ca t a ly t i c and f u l l y
c a t a l y t i c s u r f a c e s s o t h a t only a cursory d i scuss ion w i l l be presented.
Assuming a b ina ry a i r model, r e f e rence 2 a l s o p re sen t s g e n e r a l nonequil ibrium
s o l u t i o n s f o r s u r f a c e s of a r b i t r a r y c a t a l y c i t y . Two d i f f i c u l t i e s occur when
a r b i t r a r y c a t a l y c i t i e s are conce~ned . F i r s t , t h e r e is very l i t t l e d a t a on
the dependence of s u r f a c e c a t a l y c i t y on temperature and second, t h e r e i s no
informat ion t o adequately def ine c a t a l y t i c e f f i c i e n c y f o r gases wi th more
than one d i s s o c i a t e 3 spec ie . Iiowever, an important conclusior, t o be reached
from exanunation of these analyses is t h a t f o r s t agna t ion p o i n t flows t o su r -
faces of i n f i n i t e c a t a l y c i t y a t temperatures of i n t e r e s t t h e r e is only a
n e g l i g i b l e d i f f e r e n c e i n h e a t t r a n s f e r between assumptions of equ i l i b r ium and
nonequil ibrium chemist-ry . This conclusion i s e s p e c i a l l y inp0rtm.t necause of
t h e simple c o r r e l a t i o n so lu t ions which have been ob ta ined for equ i l i b r ium flows.
Thus, s i n c e experimental evidence s h w s t h a t m e t a l l i c s u r f a c e s tend t o be ca ta -
l y t i c , t h e s e equi l ib r ium chemistry s o l u t i o n s can be used t o p r e d i c t t h e h e a t
t r a n s f e r rates. I n add i t i on , f o r low c a t a l y c i t y s u r f aces , t he se p r e d i c t i o n s
r ep re sen t an upper bound on t h e expected h e a t i n g r a t e s .
The accuracy of the above analyses have been adequately proven f o r both
ground test and f l i g h t hardware, b u t i n o rde r t o apply them to s h u t t l e whose
s t agna t ion p o i n t may be n e i t h e r axisymmetric no r p l ane r , some method such as
r e f e rence 12 must be used t o account f o r t h r e e dimensional e f f e c t s . For i n -
s t ance , f o r arr axisymmetric s t agna t ion p o i n t i n a d i s s o c i a t i n g f lok 2
where 0.52 equ i l i b r ium chemistry
0. b 3 f rozen chemistry
6
Then f o r $ and RC a s t h e p r i n c i p a l r a d i i of curva ture of a non-axisynunetric
s t agna t ion p o i n t , Equation (3-1) is sca l ed by t h e express ion
where
and [NUJ~RX] is t h e s t agna t ion po in t va lue f o r an axisymmetric nose r ad ius R~ equa l t o Pi. Note t h a t f o r RC >> % t he s o l u t i o n degenerates t o t h a t f o r a
s t a g n a t i c n l i n e .
The lead ing edges of t h e d e l t a wing p o r t i o n of the v e h i c l e are no t s t ag -
n a t i o n l i n e s and should be considered as swept o r yawed cy l inde r s . So lu t ions
f o r f i n i t e l eng th cy l inde r s 'are n o t a v a i l a b l e s i n c e t h i s comprises a t r u e t h ree -
d i m n s i o n a l bo6l. I n add i t i on , nonequil ibrium chemistry has n o t been gene ra l ly
considered. Even SO, f o r c a t a l y t i c s u r f aces, adequate p r e d i c t i o n s can be made
by assuming t h a t the lead ing edge is an i n f i n i t e swept cy l inde r . Then, draw-
i n g upon s t agna t ion p o i n t exper ience an equ i l i b r ium chemistry c o r r e l a t i o n solu-
t i o n such as Reference 6 can be used. This w i l l be d i scussed i n Sec t ion 3.2.2.
Overa l l , i t i s gene ra l ly conzluded t h a t , wi th the except ion of sur faces ,
of f i n i t e c a t a l y c i t i e s , t h e confidence l e v e l is q u i t e high, even f o r v iscous
shock l a y e r flows, f o r the p r e d i c t i o n of s t a g n a t i o n and l ead ing edge h e a t
t r a n s f e r . 3 2 DOWNSTREAM RI?,GIONS
Although i t may be necessary t o cons ider viscous shock l a y e r s i n t h e
v i c i n i t y of t h e s t agna t ion p o i n t , t he major po r t i on of t h e lower s u r f a c e of
s h u t t l e can be characterize^ as having an i n v i s c i d region and a boundary
l a y e r region. Uxider these condi t ions it is t h e boundary l a y e r t h a t is of
primary i n t e r e s t , s i n 7 e an adequate a n a l y s i s of it would provide r equ i r ed
design hea t ing rates. However, t he i n v i s c i d flow, e i t h e r i n a coupled o r
independent sense , is important because it provides t h e boundary condi t ions
for t h e boundary l a y e r equat ions .
3.2.1 INVISCID FLOW
There a r e p re sen t ly some solutions13n20 f o r the i n v i s c i d flow f i e l d
about symmetric bodies a t angles of a t t a c k b u t a gene ra l s o l u t i o n f o r an a r b i -
t r a r y three-dimensional body i s n o t r e a d i l y a v a ~ table, al though some methods 21,22
are i n t h e development state. For a parametr ic s tudy of e n t r y t r a j e c t o r i e s ,
t h e i n v i s c i d flow f i e l d o r boundary l a y e r edge cond i t i ons should be determined
by approximate means; more exac t methods being reserved f o r f i n a l t r a j e c t o r y
c a l c u l a t i o n s . With t h e i n v i s c i d f low f i e l d and the boundary l a y e r assumed t o
b e uncoupled, approximate techniques can be used t o s p e c i f y t h e boundary l a y e r
edge condi t ions and a s o l u t i o n of t h e i n v i s c i d f i e l d becomes unnecessary.
It has been noted 23,24 t h a t p re s su re d i s t r i b u t i o n s a r e r e l a t i v e l y
i n s e n s i t i v e t o chemistry and d i s p l a c e r e n t e f f e c t s a t s u f f i c i e n t l y high
Reynolds numbers s o t h a t most approximation methods c e n t e r f i r s t upon ob ta in-
i n g t h e su r f ace p re s su re d i s t r i b u t i o n and then us ing it t o determine t h e chemi-
cal, thermodynamic and dynamic s t a t e s of t h e f l u i d a t t h e edge of t h e boundary layer . For simple axisymmetric bodies , i t has been shown 25 t h a t a modified
Newtonian pressure d i s t r i b u t i o n is s u f f i c i e n t (wi thin 5% of exper imental d a t a )
and i s the assumption used by s e v e r a l i n v e s t i g a t o r s 10~11.25.26. However
f o r wind tunne l d e l t a winged models, t h e modified Newtonian pressure was found 27
to be 1 5 t o 20% lower t han experimental measurements. For t h i s configura-
t i o n , Marvin e t a 1 show t h a t t h e e l l i p t i c - c o n e technique is more accurate .
However, F'annelopJo shows t h a t a s i m i l a r b u t s impler procedure ( e f f e c t i v e cone
technique) is adequate f o r s y m m t r i c bodies a t angle of a t t a c k by comparing
p r e d i c t i o n s wi th t h e exper imental d a t a of clearyS1. Laminar h e a t t r a n s f e r
r a t e s are approximatley p ropor t iona l t o t h e square r o o t of t h e p re s su re s o t h a t
r e l a t i v e e r r o r s i n h e a t t r a n s f e r rates, due t o i naccu ra t e p re s su re s , w i l l be
less than the r e l a t i v e pressure e r r o r s . Thus, p ressure approximations which
are accura te t o t h e o rde r of a few percen t should be s u f f i c i e n t and more
r e f ined approximations would be of second o r d e r and would probably be over-
shadowed by t h e ques t ion of boundary l a y e r edge chemistry. An except ion t o
t h e above would be f o r f l i g h t condi t ions f o r which t u r b u l e n t flow i s expected.
T rans i t i on phenomena i s dependent on t h e edge Mach number which i n t u r n depends
on t h e pressure d i s t r i b u t i o n . Experimentally determined boundary l a y e r edge
Mach numbers on a straight-winged o r b i t e r model were compared with those pre-
dicted from a tangent cone pressure d i s t r i bu t . i on i n re fe rence 32 and found t o
be i n good agreement. I t then appears t h a t a tangent o r e f f e c t i v e cone approx-
imation f o r p ressures should be s u 5 f i c i e n t f o r s h u t t l e c e n t e r l i n e p red ic t i ons .
There is i n s u f f i c i e n t d a t a t o reach any conclusions about p ressure d i s t r i b u -
t i o n s on the winged po r t i on of the o r b i t e r although here aga in tangent cone
o r , more probably, t angent wedge approximations should be adequate.
Given a p re s su re d i s t r i b u t i o n on t h e v e h i c l e c e n t e r l i n e , one poss ib l e
approach f o r determining t h e edge cond i t i ons would be t o c a l c u l a t e an i s e n t r o p i c
expansion of t h e f low from t h e s t agna t ion po in t t o t h e l o c a l body pressure. The
chemistry may be considered a s being i n equi l ib r ium, nonequil ibrium o r f rozen ,
wi th t h e choice depending on f l i g h t condi t ions . For a s i g n i f i c a n t p o r t i o n of
t h e s h u t t l e t r a j e c t o r y t h e a l t i t u d e s w i l l be high enough s o t h a t recombination
rates w i l l be r e l a t i v e l y s l o w , hence t h e expansion w i l l be i n chemical nonequi-
l ibr ium. Even s o , an equi l ib r ium expansion i s o f t e n used s i n c e , f o r c a t a l y t i c
su r f aces , it is known t h a t s t a g n a t i o n p o i n t hea t ing i s r e l a t i v e l y i n s e n s i t i v e
to chemistry, and t h e same i s presumed t o be t r u e elsewhere on t h e body. Fur-
thermore, equ i l ib r ium c a l c u l a t i o n s a r e s impler and edge chemist ry does no t d i -
r e c t l y e n t e r i n t o some of t h e h e a t t r a n s f e r p r e d i c t i o n schemes. 5,33-39 B u t ,
f o r onc catalytic su r f aces , nonequil ibrium expansions f o r sphere cones a t zero
incidence were suggested by ~ l o t t n e r ~ ' and were shown t o be of s i g n i f i c a n t i m -
portance by ~ e w i s ~ ~ f o r a hyperboloid under f l i g h t cond i t i ons which s imula te
po in t s on a t y p i c a l s h u t t l e t r a j e c t o r y .
Even a nonequil ibrium a d i a b a t i c expansion may be inadequate f o r s h u t t l e
because of i t s l a r g s s i z e . .Far downs t rem of t h e s t a g n a t i o n p o i n t , t h e bound-
dary l aye r edge gas does n o t o r i g i n a t e from the normal shock; r a t h e r , it
o r i g i n a t e s from an obl ique shock. A similar s i t u a t i o n a r i s e s f o r flows with
l a r g e shock curva ture s i n c e an expansion, of any k ind , from t h e s t agna t ion
region presumes t h a t the entropy a t the edge of t h e boundary l a y e r has t h e
high value generated by a normal shock wave. I n r e a l i t y , a t a s u f f i c i e n t l y
f a r downstream loca t ion t h e s t agna t ion entropy w i l l be gradual ly swallowed
by the boundary l a y e r and t h e edge entropy w i l l decrease and approach t h e
value behind an oblique shock4'. ~ a m i l t o n ~ ~ compares t h e assumptions of an
edge condi t ion determined by expansion from the s t a g n a t i o n p o i n t and t h e s t a t e
determined by assuming that the f low has j u s t passed through an obl ique shock
of s u f f i c i e n t s t r e n g t h t o g e t t h e l o c a l p ressure . As might be expected, t h e
lower edge entropy f o r the ob l ique ~ h o d c a s s u m p t i ~ n was shown t o cause an
inc rease i n t h e p red ic t ed hea t ing r a t e s .
Because of t h e e f f e c t s induced by shock curva ture and entropy swallow-
ing, a complete v o r t i c i t y i n t e r a c t i o n ana lys i s wouid be h igh ly d e s i r a b l e , a l b e i t very d i f f i c u l t . Because of t he se d i f f i c u l t i e s some semi-coupled
approximate techniques have been developed. For example, s dams 43 determined
the shock shape and pressure d i s t r i b u t i o n i n t he i n v i s c i d flow using an equi-
l ib r ium chemistry model and t h e methods descr ibed by Lomax and ~ n o u ~ e ~ ~ . Then,
assuming t h a t p re s su re s and shock shape a r e i n s e n s i t i v e t o chemistry, t h e non-
equi l ib r ium s t a t e was determined by i n t e g r a t i o n along streamtubes us ing t h i s
predetermined p re s su re d i s t r i b u t i o n . F i n a l l y , boundary l a y e r s o l u t i o n s were
obtained by i t e r a t i o n of t h e mass flow with in the boundary l a y e r and t h e
absorpt ion of i n v i s c i d s t reamtubes as pe r ~ a ~ 1 a . n ~ ' . I t becomes apparent t h a t , even approximate coupl ing techniques are very complicated and probably
very time consuming and app l i ca t ion t o more complex bodies is hampered s t i l l
f u r t h e r by t h e need f o r an adequate d e s c r i p t i o n of a three-dimensional entropy
layer .
A more exac t accounting of shock curva ture and nonequil ibrium chemistry
e f f e c t s r e q u i r e s very ex t ens ive computer codes which are not a v a i l a b l e f o r
genera l three-dimensional bodies . However, codes are a v a i l a b l e f o r bodies i n
chemically equi l ib r ium flows 14'17 bu t even under t h e s e cond i t i ons CDC 6600
computational times f o r sphere-cones a t ang le of a t t a c k are i n excess of
5 minutes. 46 For nonequi l ibr iui i chemistry and smal l angles o f a t t a c k , a per- t u rba t ion technique which uses a v a i l a b l e ze ro inc idence s o l u t i o n techniques
such a s t h a t used by F'annelopJ0 could be employed. I n genera l , un less ade-
qua te s i m i l a r i t y laws a r e developed, any technique which uses "exact" s o l u t i o n s
of the i n v i s c i d flow would r equ i r e a l a r g e computer expendi ture .
For small angles o f a t t a c k and f a r downstream of t h e s t agna t ion reg ion ,
t h e boundary l a y e r edge condi t ion i s l a r g e l y determined by t h e s tate behind a
weak,oblique shock. Then t h e degree of d i s s o c i a t i o n i n t h e i n v i s c i d flow w i l l be law a-.d i d e a l gas assumptions, a long with a s p e c i f i e d p re s su re , may be suf -
f i c i e n t f o r determining the edge s t a t e . This , of course , does n o t precluda
l a r g e amounts of d i s s o c i a t i o n caused by viscoas h e a t i n g wi th in t he boundary
l a y e r . 3.2.2 BOUNDARY LAYER
With a s p e c i f i e d boundary l a y e r edge cond i t i on , s e v e r a l approximate
methods may be used t o p r e d i c t t h e h e a t t r a n s f e r r a t e s . These inc lude Ecke r t ' s
re fe rence enthalpy 37 38 ; Reynolds analogy 4 7 t 4 8 ; rho-mu 49'50; o r swept and rnodi-
f i e d swept c y l i n d e r methods and a r e r e p r e s e n t a t i v r of methods developed by a
combination of theory , experiments and i n t u i t i o n .
Ecke r t ' s re fe rence enthalpy method i s based or1 t h e assumption t h a t
incompressible cons tan t p roper ty s o l u t i o n s can be used t o c a l c u l a t e compress-
i b l e flow h e a t t r a n s f e r i f p r o p e r t i e s are eva lua ted a t an appropr ia te r e f e rence
temperature o r re fe rence enthalpy. This r e f e rence en tha lpy is def ined as
A t a s t agna t ion po in t t h e recovery enthalpy i, i s approximately equa l t o t h e
edge enthalpy i s o t h a t e
For other regians
3 For laminar f lows r 'I. ,ISif and for turbulent f lows r 'I. fir. ~ y p i c a l imcompressible constant property so lu t ions are:
(s tagnat ion po int )
(stagnation l i n e )
(laminar f l a t p l a t e )
(turbule3t flat 2 l a t e )
where
In the reference enthalpy method these are then wri t ten a s
(s tagnat ion po int )
(stagnation l i n e )
Nu: - - - 0 . 3 3 2 (Pr*) 'I3 mq
(laminar f l a t p l a t e )
(turbulent f l a t p l a t e )
For dissociated air, the Prandtl number is nearly constant so that the reference
enthalpy equations can ts written in an alternate form by noting that
Using the experience of the analytic studies performed in Reference 2, the lami-
nar solutions can be corrected for dissociated gases and non-unity Lewis numbers
by multiplying the right hand sides by the function
where
I 0.52 equilibrium chemistry
n = 0.63 frozen chemistry
0 . 5 0 couettc flow
Then, by approximating any point on the body as a stzgnztion region or a zero pressure gradient wedge/cone, the local heat transfer can be calculated.
The Reynolds analogy method assumes that the velocity and temperature
fields are proportional to obtain a simple relationship between heat transfer
and skin friction. The Reynolds analogy factor is then defined as
The successful use of the Reynolds analogy depends on an adequate solution ?or
the skin friction coefficient and a method of calculating the Reynolds analogy
factor K. Some typical values of K are
Assumes Pr = 1, Me z 0
K ' Empirical determination by Colburn4*
Theoretical value determined by ~ u b e s i n ~ ~
K = Pr (H - M:) Empirical determination by Reshotko and ~uckers?
- 1 Sa* + 1 Von Kbrmbn
For laminar flow over a flat plate with constant properties, the skin friction
coefficient is
For turbulent flows Cf may be determined with methods set forth by Spalding and
Chi, 35 Somner and Short,53 van lIriestS4 or may use aethods such as rho-mu or
Eckert's reference enthalpy. The number of possible predictions obtained from
a permutation of R and C, is very large. Select combinations have been com- 4.
pared with - - experimental data by pearceS4 for flat plates and by Hopkins and ~nou~e'~ for flat plates and cones. Pearce concludes that the Spalding-Chi
correlation with von Kdrmdn's Reynolds analogy factor is best, whereas Hopkins
and Inouye conclude that the Van ~riest analysis with a Reynolds analogy factor
of 1.0 is best. These differing conclusions are typical for turbulent flows
and are indicative of the fact that turbulence is not a well understood phe-
nomena and the results are highly dependent on the particular application.
The swept cylinder methods are based on the analyses of Beck~ith,~ Beck-
with and ohe en, 33 and Beckwith and ~alla~her. 5 6 An important conclusion drawn
from these solutions is that, even for relatively large sweep angles, the ratio
of local heat transfer coefficient to leading edge neat transfer coefficient
along a plane perpendicular to the leading edge of a yawed circular cylinder is insensitive to the angle of yaw. For wall temperatures at which there is
no dissociation the ratio of heat transfer coefficj.t?nts based on enthalpy can
be expressed as
where
anfi 9 ' and 9 ' can be determined from tabulated resvilts presented in Reference W w,s
33. The local value of heat trans2er is then calculated from
with
and
The rho-mu method5' is based on integral solutions of the momentum and
energy equations and uses boundary layer thickness parameters and a reference density-viscosity as variable functions. These ft~nctions were determined from
available exact similarity solutions and, with suitable modifications, account
for cross-flow gradients, streamwise gradients, nose bluntness and real gas
effects. Hand calculation of the Fnt of rho-mu equations, though possible,
is not practical. To facilitate hand calcr~lations, simplified correlation
appro-.;.nations are presented in References 50 and 57 for plates, cones, swept
cylinders and the centerlines of sharp delta. In addition, modification pro-
cedures for streamline divergence and variable wall temperature cases were
described.
SECTION 4
CURRENT METHODOLOGY
Current state-of-the-art for predicting shuttle heating rates is not up
to the task of considering the vehicle as an entit) because of it^ complex
shape. In ljeu of this capability, the vehicle is usually segmented into sec-
tions for which the heating rates are separately analyzed or approximated.
Typical surface regions that are considered are shown i.n Figure 1. The stagna-
tion region, leading edges and windward surfaces have received the most atten-
tion since these regions are expected to be subjected to the highest heating
rates and, fortunately, are ecsier to approximate than the leeward side. How-
ever, due to an interaction between separation-reattachment phenomena and fran-
sition to turbulence, certain portions of the lee side have been noted to have
significant heating rates.
4.1 WIXDWARD SURFACE
Even thaugh Marvin et. al. show that Newtonia? approximations are inade-
quate for the NAR delta-wing model, the approximation is often favores because
it can be expressed in a convenient an&:.: ,. '9rm. Moreover, Newtonian approxi-
mations axe known to be adequate for sphere-cones and other simple blunt bodies
and are apparently also adequate for E f t and drag calculations of delta vehi-
cles. 58 However, one would expect that, since Newtonian approximations (based
on body angle) predict tht same pressures for cones and wedges, they would be
inadequate for regions far from the stagnation point. Hence, depending on the
region of interest, different assumptions nay be required. Young et. al., 5 9
believing that transition to turbulent flow is enhacced by entropy liyer swallow-
ing, use a high entropy Newtonian approximation in the laninar flow region and
the lower entropy tangent wedge approximation in the turbulent flow region. A
tangent wedge approximation was also used in Refere~ce 60 but boundary layer
edge approximations were modified with an empirical gas constant. Guard a ~ d
schultz61 based their pressures on a combination of experimental data and blast
wave theory of creagerG2 with a correction for nose bluntness. horna as^^ takes a novel approach; he uses a heat transfer prediction technique and model center-
line heating data and vorks backwards to obtain the prcssu-:e distribution which
yields the best correlation. With this technique he shows that blast wave cor-
relations with nose bluntness effects are best at slightly negative angles of
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attack, ~odified oblique shocks are best at zero incidence and a modified
oblique shock accounting for streamline divergence is best at slight positive
angles of attack.
AZthough approximate p~ diction methods have shown generally good agree-
ment with low enthalpy wind tunnel data, questions dealing with the effects of
chemistry, shock curvature and flow interactions need yet to be adequately re-
solved for flight vehicle prediction confidence. Even for wind tunnel models,
pressures along the windward centerline are the most often measured so that more
effort needs to be expended on pressure measuremeats on the remainder of the
vehicle.
Given a pressure distribution, an isentropic expansion or oblique shock
conditions are the simplest methods for approximating the boundary layer edge
flow conditions. However, in efforts to obtain improved approximations, some
investigators have employed more sophisticated methods to account for three-
dimensional cffeets.
Marvin et.al. assume that the edge conditions on the vehicle centerline
are the same as those behind'a swept cylinder inclined at an angle equal to the
local body incidence. However, on the wing portion of the vehicle the edge con-
ditions are assumed to be determined by an isentropic expansion from the leadin?
edge. Young et.al. use an isentropic expansion for low angles of attack whereas
at high angles of attack they use a swept cylinder theory but correct the stag-
nation line velocity gradient to account for noncircular cross sections. Masek 64
and pearceS4 account for the stagnatio? line cross flow by assuming that the
cross-flow velocity gradient is equal to that which occurs on a circular disk
with a radius equal to the local wing semi-span and a normal velocity equal to
the local normal component of the free stream flow. Pearce points out that
the cross flow correction to a plane oblique shock varies from 0 to 13 percent
for incideraces between 0 and So0 which, at least for high angles of attack,
makes the correction significant. The data and heating predictions of Marvin et.al and Guard and Schultz represent most of the reported windward side, off
centerline heating data for delta vehicles so that not much can be reported on
methods applied to the wing portion for determining boundary layer edge condi-
tions. Whereas, Marvin et.al. predict a three-dimensional edge condition, as
noted above, Guerd and Schultz do not. Guard and Schultz use a planar oblique
shock for edge conditions and correct the heat transfer results to account for
cross flow.
Current methods for predicting windward-centerline, laminar heat transfer
rates ara mostly based on adaptations of correlation methods such as Eckert's
reference enthalpy or rho-mu theories. ~ o o t e , 65 for instance, used the
r e f e rence enthalpy method bu t modified it f o r con ica l flows. A t low angles of
a t t a c k Young c t . a l . a l s o used t h e r e f e rence enthalpy methods bu t at. high ang le s
of a t t a c k , a swept cy l inde r theory (wh ich ' i s a s p h e r i c a l s t a g n a t i o n po in t theory
modified f o r two dimensional i ty and sweep) w a s used. Marvin et .al . used t h e
Beckwith and Cohen c r o s s flow theory t o modify t h e f i n i t e d i f f e r e n c e scheme of
Marvin and ~ h e a f f e r ~ ~ f o r streamline divergence e f f e c t s . This was found tc be
i n good agreement wi th d a t a f o r regions a f t of t h e wing-body junc t ion b u t f o r -
ward of t h i s p o i n t , swept c y l i n d e r theory was'found to r ep re sen t t h e d a t a b e t t e r .
For laminar f low on t h e wing of t h e v e h i c l e , t h e c r o s s f low theory of
Beckwith, and Beckwith and Cohen were used by Marvin et .al . and Young et.al.
For l i f t i n g body conf igura t ions , Guard and Schul tz used a similar a ~ p r o a c h f o r
t h e underside hea t ing whereas Reference 60 uses Ecker t ' s r e f e rence enthalpy
method and f l a t p l a t e s o l u t i o n s wi th a s 7 - z i p theory c o r r e c t i o n f o r c r o s s flow.
?or t u r b u l e n t flow along t h e v e h i c l e c e n t e r l i n e , ~ a r n i l t o n ~ ' used t h e
Ecker t r e f e rence enthalpy method with an o r i g i n f o r t u r b c l e n t f low which i s
3 s s i ~ ~ c d tc cc"zr;r zt the start oZ t i - a i b s i i i u ~ ~ . Tilt: Spaiding-Chi method was used
f o r s imulated s h u t t l e vehicl-es a t low a n g l e s of a t t a c k by Young e t al . , Masek
and ~ o r n e ~ ~ ' ] and Moote. The l as t modified t h e method used t o account f o r coni-
ca l flow and real q a s e f f e c t s by us ing a n empi r i ca l ly determined weighting fac-
tor of 1.25. A t high ang le s of a t t a c k Young e t a l . chose t o use t h e r e s u l t s of
Beckwith and Gal lagher , whereas Marvin e t al . e l e c t e d n o t t o be s e l e c t i v e and
found good agreement between t h e i r d a t a and t h e t h e o r i e s of Spalding-Chi, Sommer-
Shor t and Van D r i e s t f o r Reynolds analogy f a c t o r s of 1.0. Hopkins and Inouye
recommend Van D r i e s t ' s method wi th a f a c t o r of 1.0 bu t no t e t h a t t h e Spalding-
Chi method wi th a f a c t o r o f 1.2 a l s o c o r r e l a t e s w e l l wi th da t a .
Turbulent flow on t h e lead ing edge w a s p r ed i c t ed by Guard and Schul tz
by represen t ing it as an i s o l a t e d swept c y l i n d e r and applying t h e r e s u l t s of Beckwith and Gallagher. Turbulent hea t ing throughout t h e windward s i d e of a
l i f t i n g body w a s p r ed i c t ed by Reference 60 us ing Ecke r t ' s r e f e rence en tha lpy
method wi th a modified Reynolds analogy and t h e ~chultz-=runow6* s k i n f r i c t i o n
l a w .
It is expected i n both laminar and t u r b u l e n t f lows, t h a t t h e accGracy of
t h e pred ic ted hea t ing rates would be dependent on t h e accuracy of t h e s p e c i f i c a -
t i o n of t h e boundary l a y e r edge cond i t i on b u t Young et.al. and Pearce p o i n t o u t
t h a t t h e t r a n s i t i o n c r i t e r i a is a l s o dependent on t h e edge condi t ion . Thus wind
tunne l f r e e stream cond i t i ons may m u s s t r a n s i t i o n to occur earlier on models
t han might be expected i n f r e e f l i g h t so t h a t conse rva t ive e s t ima te s would be
ob ta ined by t h e d i r e c t use of t h e tunne l da t a .
4.2 LEEWARD SURFACE
For very small angles of attack, where the flow on the leeward surface .-
does not separate, the heating predictions have used procedures similar to chose
used on the windward surface. For example, Guard and Schultz predict the leeward
heating for a delta-body vehicle using two-dimensional rho-mu theory and noted
good agreement for both laminar and turbulent flows at angles of attack up to
30'. The good cr~rrelations suggest that separetion-reattachment phenomena* does
not occur in the vicinity where measurements were made. However, the data of
blaiseC9 for semi-pyramidal shapes at angles-of-attack less than 55O and the 70
data of Hefner and Whitehead for a delta-wing orbiter at angles of 20° and 40°
show that the leeward side heating may be nonuniform with regions of "peak"
heating. This nonuniform heating is spparently a result of three-dimensional
separation-reattachment phenomena complicated by transitional and turbulent
flows. Under these conditions it is generally conceded that the prediction of
ttis leeward heating, which requires estimates of pressure and edge flow condi-
tions,** is much too ambitious for current and even near-future technologies.
As a first approximation, Reference 60 suggests using tangent cone approxima-
tions when the local inclination is positive and use flat plate theory for re-
gions of zero or negative incidence. In *is way, transiticn and turbulent ef- fects can be included but separation-reattachment possibility would cast some
doubt on the validity of the predictions.
The only recourse to the 2kDve dilenma appears to be semi-empirical
techniques utilizing wind tunnel data. Although this would cause a decrease
in design confidence, the leeward heating is generally sufficiently low so as
not to be serious. P.ren so, it would not be desirable to be overconservative
in the selection of surface material since the leeward area is at least 50 per-
cent of the total vehicle area.
4 . 3 SHORTCCMINGS OF CURRENT IWTHODOLOGY
With the exception of the stagnation point, the current prediction meth-
ods are founded on modifications of solutions which do not, in a physical or
mathematical sense, completely account for the real environment to which shuttle
will be exposed. This, of course, does not imply that these procedures have
been ihadequate for preliminary design purposes since a large number of conditions
* Which is one of the reasons why the authors chose the delta-body rather than a delta-wing vehicle.
** In passing it should be noted that even the prediction of flow fields for simple shapes such as sphere-cones at angle of attack is no simple task and is the objective of much activity.69-72
were studied. However, for final design purposes, the present methods have
certain shortcomings that must be minimized or the methods must be replaced by
more adequate analyses. A complete inviscid/viscous flow field analysis would be
ideal but such a program is not available and its development may not be possible
with the resources allotted to shuttle development.
As already discussed, the prediction procedure for the windward side can
be considered in three different parts. First, the pressure distribution is de-
termined, then the boundary layer edge conditions are calculated, and finally an
appropriate boundary layer solution is used t~ predict the heating rates. Methods
for the prediction of surface pressures are believed,on the whole, to be suffi-
ciently accurate. The other two parts are inadequate for certain critical por-
tions of the shuttle trajectory. In establishing the edge conditions, the two
most common assumptions are that the flow originated either from a stagnation
region or an oblique shock. The former is valid near and the latter far from
the stagnation region. dams^^ shows that "far" may be as high as 15 nose radii d3;:nstrean for sphere-cone Sdies. In between these extremes, say between 3 and
15 nose radii, a three-dimensional entropy layer will form and swallowing by the
boundary layer may be important. None of the above methods accounts for this
entropy swallowing effect. In addition, present methods rely on solutions ob-
tained with similar and locally similar assumptions when in fact the boundary
layer will be highly nonsimilar.
A somewhat more serious question arises fron flow field chemistry. The
present methodology makes use of analyses developed for ideal gas flows and cor-
rects for real gas effects by empirical methods ( e . g . , Eckertls reference en-
thalpy method), Although these precedures have been shown to yield good agree-
ment for stagnation flows with equilibrium chemistry, their application to down-
stream flows of complex bodies is not fully justified and in nonequilibrium
chemistry with noncatalytic surfaces, these methods fail except to bracket the
probable heating range, Moreovzr the prediction methods are based on similarity
or local similarity assumptions which are generally invalid for nonequilibrium
chemistry boundary layers.
Turbulent boundary layer heat transfer predictions may be very important
for shuttle. As noted though, there are several prospectively good approxima-
tion methods; the choice of which may depend on the particular geometry and/or
environmental conditions. Noae of these methods is believed to be universally
correct; in fact, since none of these methods directly account for nonsimila-ity
and thennochemistry effect , they icsy be seriously in error for shuttle even though they adequately predict wind tucnel heating rates.
4.4 RECOMMENDED CURRENT STATE-OF-THE-ART METHODOLOGY
It has been noted that pressure distributions are relatively insensitive
to chemistry so that pressures can be determined from ideal gas flows in the
form of experimental correlations or analyses. Modified Newtonian flow methods
are the simplest to use since the pressure can be expressed analytically, but
it was shown in wind tunnel measurements on a scale delta-wing vehicle that the
predicted centerline pressures are about 15-20 percent lower than measured val-
ues. Nonetheless, for rough-cut calculations on the vehicle centerline, this
should be sufficient. For more accurate predictions a tangent cone method is
recommended over the more complex elliptic cone method. A comparison of these
two predictions and the data of Marvin et.al. is shown in Figure 2 and good
agreement is obtained up to an angle-of-attack of 40°. At ~ 3 . 5 ~ the tangent
cone method overpredicts the pressure by about 10 percent whereas the elliptic
cone method underpredicts by about 5 percent. Thus the tangent cone approxima-
tion will lead to conservative (higher) heat transfer predictions. In retro-
spect, the failure of the tangent cone approximation at 53.5O shoilld be expected
since this angle approaches the maxi.num c<;ile angle that wili support an attached
shock and, at these high andes, the precedure would begin to fail even for
slightly blunted cones.
If an approximate analytic equation is desired, a modified tangent cone/
wedge method can be developed as follows. The Newtonian pressure is given as
where a is the angle of the surface with respect to the free stream vector. At
a stagnation point the Newtonian pressure is higher than the actual pressure
whereas on sharp wedges and sharp cones it is lower. The mcdified Newtonian
pressure adjusts the predicted value to be correct at the stagnation point;
then the pressure is given by
This modified Newtonian pressures does a very good job of predicting the pres-
sure distribution on the forward portion of blunt bodies, but far from the nose
recion this procedure predicts a slightly lower pressure than the unmodified
method. Thus the discrepancy is widened further. Since the Newtonian formula-
tion is analytically convenient the following procedure is recommended for large
ratios of surface distance to nose radius. Let
where pr is the pressure on a reference surface oriented at the angle ar. This
equation can be written in the form
which is then a Newtonian tangent cone/wedge equation. To illustrate the use
of this equation, cqnsider an airfoil at angle of attack as shown below
The chord line will be used as a, and, depending on the nature of the flow around
the airfoil and whatever body is attached to it, the pressure p can be deterained r for a wedge or cone at angle ar. The pressure at A which has a local angle of'
incidence equal to a can then be calculated from Equation (4-1). For a typical
shuttle vehicle this procedure will yield pressures comparable to tangent wedge
or tangent cone values as shown in Figure 3. Ncte, hoazver, that a qualitative
decision must still be made regarding which, tangent cone or tangent wedge,
is most applicable. Note also that the procedure would not be valid in the
limit as ar approaches zero.
At low angles-of-attack the transverse curvature of the shock wave on
the windward side will not be significant except near the centerline of the vehicle.
The flow across the wing is then akin to that around a yawed-blunted wedge so
that the pressures should be adequately predicted with tangent wedge approximations.
At higher angles-of-attack the shock wave, even over the wing, will have signifi-
cant transverse curvature so that tangent cone approximations should be used.
It is apparent then, that a decision must be made as to when each assumption is
valid. Based on the Marvin data, the following rule-of-thumb is recommended:
If the angle-of-attack is somewhat greater than the half-angle of the delta,
then use the tangent cone approximation; and if the angle-of-attack is approxi-
mately equal to or less than the half-angle, use the tangent wedge approximation.
Regardless of which approximation i s used on t h e wing, t h e co ro ina t ion
of t h e a i r f o i l shape and d i h e d r a l causes t h e t r u e l o c a l ang le of incidence t o
be d i f f e r e n t from t h e v e h i c l e angle-of-at tack. For smal l va lues of t h e sum
(i + g ) , a geometric a n a l y s i s w i l l y i e l d t h e fol lowing r e s u l t s f o r chordwise
s t a t i o n s g r e a t e r than a few percen t .
s i n y = cos 4 s i n ( a + ?i + E )
where
a = veh ic l e r e f e rence l i n e of a t t a c k - u = angle between chord l i n e and r e f e rence l i n e - - a = l o c a l s u r f a c e i n c l i n a t i o n wi th r e s p e c t t o chord l i n e
$ = l o c a l wing d i h e d r a l ang le
y = t r u e angle of incidence on wing
Calcu la t ions us iug y i n t h e t angen t wedge approximations a r e compared with t h e
Marvin d a t a i n Figure 4. It should be noted t h a t t h e gene ra l decrease i n preP-
s u r e going outboard i s due p r imar i ly t o a change i n r a t h e r than a three-dimcn-
s i o n a l end flow e f f e c t .
The f i n s can be handled i n much t h e sane way a s t h e wings where once
aga in t h e t r u e inc idence of t h e f i n n u s t be determined, Assuming t h a t t h e f i n
i s a f l a t p l a t e wi th no a i r f o i l shape, t h i s incidence i s given by
s i n y l = cos $ ' s i n 2 a + t a n 2 $ ' s i n ( E + a) 7
where
-
s i n a
sin ' = c i , . , -
a = angle between Fin chord l i n e and v e h i c l e symmetry plane
= angle of tilt of f i n
Y ' = t r u e angle of incidence
There is no a v a i l a b l e d a t a on f i n p re s su re d i s t r i b u t i o n s t o use f o r camparisons
of p red ic t ed values .
The pressure along t h e s t agna t ion l i n e of both t h e f i n and t h e wings
are c a l c ~ a l a t e d as t h e p re s su re on a swept c y l i n d e r wi th a c o r r e c t i o n f o r angle-
of-attack to obtain the true yaw ansm.e. That is, for the wing, with a small
dihedral angle
cos A = cos a cos C
where
= semi-apex of wing
A = true angle of wing leading ledge
and for the leading edge of the fins
where
cos A ' = r sin(& + a)
r -;I -1 , I 2 r = 1 + (sin 6 tan a + cos 6 tirn 4 ' )
-I J
6 = nominal sweep angle of fin measured on vehicle symmetry plane
A' = true angle of attack of fin leading edge
The 'above angles fcr the wing and fin are shown in Figure 5 and the above rala-
tionships are derived in Appendix 1.
As far as the leeward side is concerned, short of a complete Elow field
analysis, there are no adequate means of predicting surface pressures. Based
on the analysis of Reference 73 and experimental data,74 it is rtcommended
that laminar heating rates be calculated as 0.56 of the laminar flat plate
value and turbulent rates at 0.84 of the turbulent flat plate value.
For the lot7 velocity-low altitude portion of the shuttle trajectory.
the predicted pressure can be used with equilibrium gas assumptions to obtain
adequate boundary layer edge conditions. But during the earlier portions of
the trajectory, accurate preZict,io:is of the edge condition must include the
effects of nonequilibriurr. chemistry, especially if surface kinetics cre to be considered. However, fol a "fi-st cut" design calculation, equilibrium
chemistry can be assumed with the knowledge that the use of noncata,,ytic surfaces
affect the heating rates both at the non-catalytic surface and downstream of it.
For a distance up to several nose radii do:~ns~tream, the edge condition can be
determined by a streamline expansion.
OUTBOARD F I N NOMENCLATURe
F I G U ~ E 5 BODY REFERENCE A N A L Y S I S
Far downstream (say a distance in the order of tens of nose radii) the edge
condition can be assumed to be the sane as that which would exist behind an
oblique shock with p2 equal to the local surface pressure. Thus with p2/p1 * kn3wn, equilibrium normal shock tables can be used to determine the equivalent
free stream Mach ni~nber, Mequiv. ' which is equal to Moosin 8. The tables will
also yield all static properties and the normal component of velocity; thus
with the tangential terms unchanged the edge velocity can be caiculated to
complete the edge conditions. For intermediate distances, there will be a
transition from the high entropy stagnation state to the low entropy oblique
shock state. The rate of transition depends on the shock wave shape and is
further complicated by three dimensional effects. Since the heat transfer
prediction methods to be recommended use similarity or local similarity assump-
tions, the results are not sensitive to the smoothness of the edge conditions.
Thus it is suggested that, for moderate distances from the stagnation point,
the solution be bracketed by solving both the normal shock expansion and the
oblique shock conditions.
It is believed that the particular choice of an approximation method
to be used to predict the heat transfer rates is not critic21 since a
degree of enpiricism is built iato each method. In line with the tangent
cone/wedge method used t~ predict pressures, it is recommended that the heat
transfer be predicted using, for example, Eckert's reference enthalpy method
for conical or wedge flow where the local body incidence is useC! as the cone
or wedge angle. This procedure, of course, would not be valid near the nose
or leading edge but would be applicable at large S/R. Calculations, as described
are compared with the Marvin's centerline data for a = 15O and a = 30° in
Figures 6 and 7 respectively. The 53.5O case is not compared since, as noted,
this is too close to the limiting cone angle. The agreement at a = 30' is very
good for X/L - > 0.2 but at a = lSO the predicted values are conservatively higher
than measured values.
Stagnation point heat transfer can also be calculated using the reference
enthalpy method but between the stagnation point and x/L % 0.2 the method of
~ e e s ~ ~ is suggested. Briefly, the ratio of local heat tlansfer to stagnation
heat transfer is given as
* e.g., Reference 75.
where
0 two-dimensional
1 axisymmetric
and the coordinates are defined by the following sketch:
An alternative approach for regions not too close to the nose is that
used by Guard and Schultz. On the windward centerline this procedure calculates
the two-dimensional heating rate and scales it with methods developed by
~ u n a v a n t ~ ~ and Thomas, et. al. 57 For example, the local heating can be predicted
by Eckert's method for flow over a plate or wedge, i.e.,
Nu: 1
- = 0.332 (Pr*) 3
then the ratio of heat trasnfer coefficients is
(e) =
Lam
where
a
2 ( tan ~)(M=cos a
and E is the half angle of the delta wing, The above procedure would be valid
only for the centerline of the delta portion of the vehicle. When turbulent
flow is exiected, it is observed from Equation (4-3) that the effect of stream-
line divergence is very small and can probably be neglected.
For some vehicle configurations, (e.g., NAR configurations 129 and 134)
the delta portion is preceded by a cylindrical fuselage. In this region, the .
swept cylinder theory of Beckwith, snd Beckwith and Gallagher should be used.
On the wing and at low angles-of-attack, two dimensional methods are probably
sufficient but at higher angles, cross flow effects should be iccli~ded following
the procedure of Beckwith and Cohen.
To be conservative, it is recommended that "spot" calculations be made
to determine upper and/or lower bounds in terms of 2D/3D and laminar/turbulent
predictions. This of course would not be necessary if exact solutions were
available. In addition the degree of confidence in the overall predictions
is increased as exact methods for local predictions become available. The
recommended procedures, which are summarized in Table 1, are not expected to be
applicable to all possible vehicle configurations but are considered adequate
for the current delta configurations. In using the recommended procedures or any other approximation scheme, it should be noted that, although a number
of possible "improvementsn are available, the increased complication is often
not warranted in view of the fact that these improvements are building onto
solutions that do not account for all of the important phenomena. The test of
the accuracy of approximation methods is a comparison with exact solutions or
flight data. Because of the budgetary and political conskraints imposed on
TABLE 1
RECOMMENDED METHd3S FOR APPROXIMATE HEAT TRANSFER
PREDICTIONS TO DELTA SHUTTLE VEHICLES
+ Heat Tran8fer
I
Fay and Riddell correlations (or Ref.
5)
(::::1*"
{l
+ [=e0ms2 - 1
)
qo - 0
.94(~~u~)~*~
- [; ,/-I"'
(io -
iw)
or Reference enthalpy
Nu*= 0.
76(~r*)~** 1 +
-
JRt*
Reference enthalpy, tangent cone
qtc
e
1 t
(~e@-''
- 1)
Swept c
yli~der - Beckwith
0.5i
A
+ (
~e
@*
~~
-
JRT
or
qsc =
0.75
qo(sin A
)"~
Tangent cone aeff >
semi-apex angle
qtc
as in region A
2,
Aj
Tangent wedge aeff
< semi-apex angle
1
qtw =
qtc
or ctossflow theory, Ref. 33,
39
Approximation from experimental data
I
Edge Conditions
Equilibrium stagnation,
isentropic expansion
Oblique shock conditions
(tangent cone
Swept cylinder stagna-
tion line, isentropic
expansion
Oblique shock conditions,
tangent cone aeff 2
semi-
apex angle
Tangent wedge aeff <
semi-apex angle
---
Region*
A1
A2'A3
B1,B2,C
D,E1
K,G,E2
Pressure
Stagnation point and
Newtonian
Tangent cone
Swept cylinder stagna-
tion line
Tangent cone for aeff '
semi-apex angle
Tangent wedge for aeff<
semi-aper. angle
Approximations based
on experimental data
See Figure 1
s h u t t l e , it i s no t p r a c t i c a l t o wait f o r f l i g h t d a t a , hence t h e development of exac t o r near-exact computer codes (a t l e a s t for s p e c i f i c reg ions of t h e
s h u t t l e su r f ace ) i s t h e key t o a succes s fu l s h u t t l e design.
SECTION 5
CANDIDATE ADVANCED BOUNDARY LAYER TECHNIQUES
During the past few years with the advent of large scale computer machinery
a number of computer codes have been developed which yield numerical solutions
to the boundary layer equations. These codes are relatively expensive to operate
and are often tempermental. For these reasons it is not ecvisioned that they
should take tke place cf experiments or simple engineering relations for predict-
ing hbzt-transfer rates when such methods are available. However, they are very
useful in auxiliary studies where they can be used to validate or calibrate
engineering relations, to extend engineering relations to include additional
effects, to extrapolate wind-tunnel tests to flight conditions, and to deveiop
correlations for use with simple engineering relations. A classic example
of this last approach is the correlation of stagnati~n point boundary layer
solutions by Fay and Riddell. 2
While boundary layer computational technology has come a long way,
no code is available (nor would it be practical to develop a code) which treats
the complete shuttle boundary layer precisely. This would require consideration
of a three-dimensional boundary layer with separated flow and viscid-inviscid
coupling over the leeward side of the vehicle. The only three-dimensional.
code available today which is oriented to flight vehicle gecmetries and which
treats the equations precisely is that of Der. 71 However, this code uses an
explicit approach and therefore is very expensive to operate. Also, it does
not consider separated flow, viscid-inviscid coupling, turbulence, or chemical
reactions.
A rather extensive review of boundary layer computatic technology
was p5esented recently in Reference 78. This report discussed in detail the
various nuxerical procedures presently in use (e.g., shoot and hunt,
quasilinearization, streamwise integration, implicit and explicit finite differ-
ence procedures, and successive approximation and Newton-Raphcon iteration
methods), It also discussed the current status of all then known codes for
treating the chemical state, coupling to inviscld flow, coupling to surface
phenomena, molecular transport, turbulent transport, and three-dimensional flows.
While a few new codes have appeared since this report was written in late 1968
and some of the codes discussed therein have been developed somewhat further, t Z
g the basic conclusions of the report are not changed. Therefore, the reader
4 i is referred to Reference 78 for a comprehensive comparison of candidate codes.
Based on the results of this survey, it would appear to be practical
to develop a code for shuttle application with the following features:
1. Consider? the full nonsimilar axisyrmnetric or planar boundary layer
equations (i.e., it contains no similarity approximations).
2. Solves the boundary layer equations "exactly1' in a n~lmerical sense
(i.e., without the use of linearization, assumed profile shapes, etc).
3. Considers laminar, transitional, and turbulent flows including rather
sophisticated models for transition length and turbulent eddy viscosity
4. Performs calculations along inviscid streamlines with approximate
treatment of three-dimensional effects through use of small cross-
flow theory (the axisymmetxic qnalogy).
5. Considers regions of attached flow only.
6. Considers nonisentropic boundary-layer edge expansions (i.e., entropy
layer sf f ects) . 7 . Considers detailed nonequilibriurn chemistry including surface-catalyzed
reactions or equilibrium chemistry when the situation dictates.
8, Permits consideration of surface ablation materials through use of
a relatively general chemistry model and general ablating wall
boundary conditions.
9. Performs each streamline solution around the body in a matter of a
few minutes on a relatively high speed computer such as the Univac 1108.
There are possibly several dozen separate codes in the country today
which perform Items 1 an? 2 above, that is, which yield "exact" numerical
solutions to various sets of boundary layer equations. However, most of these
are limiced to incompressible or compressible single-component boundary layers
or have some other major shortcoming that rules them out of consideration for
application to Shuttle. Experience in developing the BLIMP code at Aerotherm
would suggest that the development of an operational compressible single-
component code is a trivial fraction of the effort needed t . ~ develop a code
of the type needed for Shuttle.
Limiting attention, then, to reacting, nonsimilar boundary layer codes,
there are about six codes (or classes of codes) which would appear to have
some potential for application to shuttle. These are
1. Implicit method of Blottner as extended and used by several investigators
2. Shoot and hunt method of Smith
3. Implicit method of Cebeci
4. Multiple strip method of Pallone
5. Implicit met.iiod of Spalding
6. Newton-Raphson method of Kendall (BLIMP program)
Probably the earliest nonsimilar nonequilibrium boundary layer code
was developed by Blottner while at General Electric. lo In Reference 10 he ex-
tended the implicit finite difference procedure developed while a Flugge-Lotz
student7' to binary nonequilibrium (atoms-molecules) . The basic computational
approach developed under this thesis was used later by several subsequent stu-
dents including ~annelo~" and ~avis.*l The Blottner binary nonequilibrium
code was obtained by Boeing and extended by Tong to mu~ticomponent air nonequil-
ibrium.82 Blottner also extended his code to a nonequilibrium air model. 11 -
This code has been used and extended at ARO by Adams, 4 3 f 83 ~avis'~ and Lewis 85
in studies of chemical nonequilibrium, mass transfer, viscous interaction and
turbulent flow (e~uilibrium only). The General Electric version has also been
extended to include ablation products 86'87 and to apply to the thin shock
layer equations. 88
Another arly nonsimilar boundary layer code which has gotten extensive - -
use was developed by Smith and clsttera9 of McDonnell Douglas. This code
uses a snoot and hunt method. 78 It was extended to nonequilibrium air in
Reference 90 and to include binary (foreign gas; injection (but for equilibrium)
in 2eference 91. This latter version has been used, for example, by Mayne
and coworkers. 92 Further extension of these codes seems to have stoppsd with
the advent of an improved implicit finite difference metiiod by Smith and .
Cebeci . 93 '94 This code has been used rather extensively ky Cebeci (e.g.,
Ref. 95) in turbulent model studies including air-to-air injection but while
still retaining a nonreacting com?ressible boundary layer framework. Other
compressible turbulent nonreacting boundary layer codes have been developed.and
used in turbulent model studies, the most noteworthy being Bushnell and Bec~with 96
97 and Herring and Mellor.
A third early ap~roach for solving the nonequilibrium, nonsimi'ar boundary
layer equations was developed by Pallone and covorkers. '* They employed a
multiple strip integral method. 78 This c ~ d e dcrs not appear to have been used
too ~tensively, but some calculations are reported in Reference 99 considering
teflon ablation products and further solutions are promised (but not presdnted)
in Reference 100.
In 1967 Spalding and Patankar published a booklo' containing listings
end instructions for use of a skeletal computer program based on an implicit
finite difference procedure for solving chemically reacting laminar or turbulent
flows, the idea being that the user would supply the necessary subroutines
(chemistry, turhlent model, etc.) to solve his specific problems. This
method has been used extensively by Professor Kays of Stanford University and
his students in studies of the turbulent (principally Ancon~pressible) bo~nd~ry
layer. lo2 It was also used by Mayne and Adams lo3 in a study of streamline
swallowing in laminar boundary layers. No other Americ-n references could
be found. Certainly a lot of effort would be required to develop this method
to the point where nonequilibrium chemistry is included.
In 1966 a novel implicit procedure employing Newton-Raphson iteuXation 78
was developed by Kendall and Bartlett of Aerotherm. lo4 This method ha= been
used extensively to study equilibrium chemically-reacting laminar ana turbulent
boul~dary layers with and without surface ablation, including entropy layer
effects, and including the axisymrnetric analogy to three-dimensional flow.
A fairly recmt version of the program which is termed BLIMP fs described in
Rs!ference 105. While the code is currently operatioral for general equilibrium
flows only, subroutines governing homogeneous nonequilibrium axe currently
built into the code which, while not fully operational within BLIMP, are operaticnal
as a separate nonequilibrium~streamtube code.
Additional code developments worthy of men+~on include those of Galowin
and Gould of GASL, 106t1'07 Mondrzyk of ~oein~,'O* Mo~re and Lee of TRW, 109,210
Chenoweth of Sandia, and Marvin and Sheaffer of NASA Ames. 66 The first two
codes treat nonequilibrium but employ time cons~milg explicit prccedures. They
appear to have been used very sparingly. Moore and Lee present c,olutions for
discontinuous inert injeciion into a n~nequilibrium laminar bounda,ry layer
but no sol.uticns seem to have been repo~ted since their initial efforts.
Chenoweth presented plai~s for a nonequilibrium turbulent boundary layer t;or?c,
but it is not clear that the code has ever been developrd. The code of Marvin
and She;ffer, while limited to a binary equilibrium mixture, i.c, mentioned
since it is being used to evaluate shlittle heating data. 112 127
While the codes mentioned above con~.:der for the most part reacting
nonsimilar flows, they are not all ideally suited for application to shuttle.
Some apply only to sharp bodies lo8 and some require special starting procedures
(eg., marching from a known solution) . '* Many are currently limited to laminar
flow (e.g., Refs 86, 99 and 110) and little experience has been gained with
some (e.g., Refs 101, 106, 110 and 111). Many consider only air chemistry 11,90,98
While this .would be suff5cient for nonablating heat shields, it would be
severely limiting in the event that replacable ablation panels come into vogue.
Thore are several other considerations such as calculation.31 speed, ease of setting up and running problems, and ~znerality -- f i ) other words, usability.
Taking all ~f these factors into consideration it appears that t.here are two
major candidates for a snuttle b~uadary layer code -- the BLIMP cod2 and a re- cent version of the Blottner code.
Extension of the BLIMP codc would require the implementation of the hono-
geneous kinetics subroutine and other technical but strai~htforward modificatiocs
requ!..-ed to achieve nonequilibritm solutions with nominally the sane accuracy,
stalility, and coinputational speed with which equilibrium solutions are currently
generated. Exrension or a Blottner code would require (depending upon the spe-
cific starting point) the implementation of a transitional and turbulent hcating
model, addition d f the axisymmetric an.rlogy, and the addition of a general abla-
tion capibility. Judging from the number of codes which were developed in the
early 1960's to include nonequilibrium 11'90898 and the fact that turbulence and
ablaLlon phenomena have been included only recently, it would appear that the ex-
tension of BLIMP to nonequilibrium wo?lld ba substantially easier. Secondly, the
resslting code would be apt to be considerably more general and flexible since
this is the major advantage of the R L I W code over other codes today. Finally,
based on the experience of one of the authors (HT) who has used both codes, tbe
BLIMP code would be expected to be easier to use and less expensive to operate.
BLIMP PXEDICTION CAPABILITY
SECTION 6
BLIMP is a nonsimilar, chemical equil.ibrium computer code which uses a
splf ne fit tccf-,?iquc for approximaliklg the distribution of flow variables within
the boundary layer. The present capability includes an arbitrary edge pressure
distribution, surface kinetics, laminar and turbulent flows, two-dimensional
entropy layers and cross flow (axisymmetric analogy). All of the above features
have been validated either on wind tunnel models or Apollo El ight data. The
spline fit technique is very efficient in terms of computational times when
compared with finite difference methods.
In its present state, BLIMP can be used to predict pitchplane heating
rates to the windward and leeward (for no separation) surfaces during the
equilibrium chemistry portion of the trajectory. To make BLIMP applicable to
the rest of the trajectory, norlequilibrium chemistry and three-dimensional entro-
py layers should be included. Even without these modifications it is believed
that nonequilibrium heating rates to catalytic surfaces are not significantly
different from equilibricm heating rates, so that with a bit of em?iricism,
BLIMP c-uld be used for the full trajectory provided the shuttle surfaces are
catalytic.
For noncatalytic surfaces, BLIbP in its present state is icadequate
and nonequilibrium chemistry of some sort is essential. Since the primary
driving function for heat transfer is enthalpy, an elaborate multi-component
representation is probably n ~ t required. It has been shown that binary models
are sufficient in most cases, but for shuttle, since oxidation is important,
a minima model would require at least three species, namely atomic oxygen,
atomic nitrogen, and molecular "a;rn. With nonequilibrium chemistry, BLIMP
would then be able to handle surfaces with discontinuous surface catalycity and
oxidation rates.
BLIMP, like other boundary layer codes, requires a means of either
determining or specifying the edge conditions. A scheme such as that used by
Adams can be used since BLIMP'S present capability allows for streamline
absorption into the boundary layer. Alternatively, che methods described in
Section 4.4 can be used.
Since BLIMP account^ for most of the phenomena associated with entry
vehicle heating, a favorable comparison with wind tunnel test data generates
a high degree of confidence in flight predictions. The accuracy of these
flight predictions have bsen proven for Apollc =lass vehicles and this experience
suggests that a code such as BLIMP should be used in a sensitivity study and
the results compared with wind tunnel data to determine which basic assumptions
and phenomena will be of significant importance in flight. It should be stressed,
though, that wind tunnel conditions will generally not include chemical dissoci-
ation-recombination effects so that the ability of a code to adequately account
for these effects must be determined from extensive experience with th? code.
To assess the effects of various three-dimensicnai and entropy layer
assumptions, a matrix of heat transfer solutions were obtained for the windward
generator of the NAR Model 134B and compared with the wind tunnel data of
Reference 27 and 113. Additional supporting data were obtained by direct
conununications with the authors of Reference 27. The body geometry of the NAR
Model 134B was obtained from Reference 114 and shock shape was measured from
shadowgraphs of the flow about NAR Model 12g115(which is slightly larger than
Model 134B). These shadowgraphs were obtained from J. Cleary of NASA Ame.s,
Sii~ce the wind tunnel test conditions were at relatively low temi;..rcL'~res,
the homogeneous version of BLIMP which consumes substantially less copF.ter
time than the chemically reacting version was used. These short run times
make homogeneous BLIMP an economically practical tool for extensive studies
of winC tunnel test conditions. Several check runs wer.-. made vith the chemical
code to verify the accuracy of the homogeneous results and in all cases,
predicted heat fluxes were found to agree within 2%.
6.1 PRESSURE DATA FOR BLIMP INPUT
Pressure ratio data (P/Pt2) were taken directly from the pressure ratio
plots in Reference 113. The total pressure behind the shock was obtained assuming
an ideal gas with y = 1.4 and M = 7.4. Lt is noted that there is undoubtedly
some error introduced by interpolating from such a plot; however, the uncertainty
introduced into the heat flux predictions will be only about 50 percent of the
uncertainty in pressure and thus this socrce of error is considered small (no
more than 3 percent at the lowest angle of attack). For the purposes of the
BLIMP code input, additional pressure data were required, particularly in the
nose regions. These data were generated by assuming a Newtonian pressure
distributi~n from the ataqnation point and calculating the pressure ratio from
the measured slope of the windward oenerator (see Section 6.1.2). The pressure
distributions used in the predictions are given in Appendix 2 and consisted of
these computed values faired into the forwardmost reported measured values.
Pressure gradient data (a(P/PT)/as) for the spreading factor calculation were
obtained by measuring slopes froa the pressure ratio vs. s plots.
Heating data were obtained from the normalized plots in Reference 113.
in a similar manner. Actual heating values were computed using the calculated
stagnation point heat flux values listed in Table 2. These values were computed
by Marvin based on a 0,006 foot radius sphere and were used to normalize his
data, These values are not measured stagnation fluxes.
6.2 MODEL GEOKETRY
The side profile and several frontal cross sections of the NAR Model 134B
are shown in Figure 8 to provide a general indication of the model configuration.
Pertinent geometric variables are
s , distance along the windward generator at each angle of attack. Here it was assumed the stagnation point was located at the inter-
section of the windward generhtor and a radial line through the
center of the nose sphere at the particular angle of at-tack. The
body centerline was taken froin the drawing co be a line through
the center of the nose sphere and parallel to the straight sections
of the upper and lower fuselage.
R, , radius of curvarve of the windward generator
% , radius of curvature at the windward generator in the plane transverse to the generator. This plane is locally perpendicular to the
generator.
r0 the perpendicular distance from the windward generator to the
angle of attack axis, i.e., the local radius of an axisymmetric
body generated by rotation of the windward generator about the
angle of attack axis.
6.3 SHOCK WAVE GEOMETRY FOR EWROPY LAfER PREDICTIONS
Shock wave geometry was obtained from full scale shadowgraphs of Model 129
which were obtained at the same freestream conditions as used in heat transfer
and pressure tests. Angle-of-attack conditions included a = 15O, 30°, 45"
and 60°. Data for the a = 53.5' test condition were obtained from interpolation
between the a = 4S0 and 60° cases.
TAB
LE
2
TE
ST
CO
ND
ITIO
NS
FOR
T
HE
PR
ED
ICT
ED
CA
SES
* Cs
lcu
late
d b
y N
~sA
/Am
es p
erso
nn
el;
bas
ed
on a
0
.00
6
ft
ra
diu
s sp
he
re.
Re
fere
nc
e
Sta
gn
ati
on
Po
int
Hea
t F
lux
*
(~
tu/f
t2
se
c)
I
48
.54
48
.72
99
.82
44
.18
@ S
tag
na
tio
n P
oin
t
(OR
)
57
1
58
1
65
0
62
1
Wal
l T
em
pe
ratu
re
(OR
)
54
5
54
0
57
0
56
0
Ca
se
a =
15
O,
La
min
ar
a =
30
°,
La
min
ar
a =
30°
, T
urb
ule
nt
a =
5
3.s
0,
La
min
ar
+
Fre
e S
tre
am
To
tal
Pre
ss
ure
(psis
)
25
2
23
9
1,4
94
25
7
Tot
r.1
Te
mp
era
ture
(OR
)
1,4
66
1,5
00
1,4
09
1,4
31
The shock angle (given in Appendix 2) relative to the angle of attack
axis was measured as a function of the cadial distance from the angle of attack axis to the shock and then converted into i~ial pressure ratio across an oblique
shock by the relation
This ratio together with the shock radius was used for the entropy layer input
for the homogeneous code.
6.4 PPPROXIMATION OF THREE-DIMENSIONAL FLOW EFFECTS
Three dimensional effects can be approximated in the BLIMP code by an ' *
axisymmetric analogy or in terms of a streamline spreading factor h2. In
the former approximation, the heat transfer rates are calcul~ted by assuming
that the body is axisymmetric with a longitudinal profile s~ecified as the
windward generator. The continuity equation is then given by
For a spherical surface such as on the face of an Apollo vehicle,h2 is given 116
by the equation
where
d2h2 1 aP dh2 - - cos €IB h2 = 0 ds2 p,r as ds I
B B is tho angle between the local surface normal and the free
stream velocity vector
PT is the total pressure
% is the local surface radius of curvature
For this case, there is no ambiguity in the variable Rc since the surface is
spherical, but for a general chree-dimensional body such as a shuttle vehicle
" h2 is also tte metric coefficient of the streamwise coordinate
the above formulation does not account for the difference in surtace curvature
as measured in the transverse and longitudinal ciirections A new derivation,
presented in Appendix 3 , shows that for non-spherical surf~cer the spreading
factor s;lould really be defined by the equation
Equaticn (6-3) was integrated using a Runge-Kutta method for 3 single
second order equation. The dynamic pressure term was evaluated from the perfect
gas relation
pu' P~
resulting in the equation being a function of the two surface radii of curvature,
the pressure ratio, and the pressure ratio gradient which was evaluated from a
plot of the measured pressure supplemented by the calculated Newtonian pressure
gradient distribution near the nose. Calcclated values of h2 are given in
Anpendix 2.
6.5 BLIMP PREDICTION MATRIX
Experimental data were available for laminar flow at a = 15O, 30° and 53.5O and turbulent flow at a = 30°; the matrix of BLIMP solutions, shown in
Table 3, is limited tc these angles-of-attack and the test conditions shown in
Table 2.
6.6 RESULTS AND DISCUSSION OF BLIMP STUDIES
Figures 9 through 12 show the influence of various assum?tions on the
predicted heating rates. The small crossflow results shown are all based upon
P/PT replacing cos2eB in the spreading factor equation (Equation (6-3) ) and
RT = - in the aft or wing region of the fuselage. A11 predictions for a given
case used the same stations, pressure ratios, and wall te~nperatures input distri- butions. Three-point differencing was used throughout to obtain streamwise de-
rivatives except for the turbulent case which used two-point differencing.*
" A run using three-point differencing was also made with negligible difference in the results in the laminar regjf~n. Two-2oint differencing was used to provide more flexibility through the transitjon region than afforded by the limit of 4, two-point difference stzitions in the three-point difference optian.
TA
BL
E
3
MATRIX OF BLIMP PREDICTIONS
-...
-- -
--
-
Axisymmetric Blunt
Entropy Layer
a =
3
0°,
Laminar
a =
30
°,
Tu
rbu
len
t
a =
5
3.s
0,
Lam
inar
Laminar Flow Results
The blunt planar body predictions made for the a = 15* and 30' (Figures
9 and 10) laminar cases undexpredict heating over the entire length of the wind- ward generator. The differences increase with angle of attack to roughly
50% for a = 30°. These trends are consistent with the over-prediction of
boundary layer growth resulting from the planar approximation as compared to
an axisymmetric treatsect.. Althoilrjh the planar blunt body assumption may be
useful in establishing a lower limit on the predicted heat transfer rates, its
use is not justified since there is no significant reduction in time to either
set-up the input data or run the code. However, since no wing heat transfer
predictions were attempted, no conclusions can be reached on the validity of
this assumpticn in these regions.
The predictions based on an isentropic expansion around a blunt axi-
symmetric body compare favorably with the data for the a = 15' and 30'
cases (Figures 9 and lo), particularly over the aft regions of the fuselage.
In the nose anc? forward regions of the fuselage, the predictions are somewhat
low; an apparent effect of the lower degree of streamline spreading caused
by the assumed axisymmetric body as compared to that which should exist over
the smaller transverse curvature of the actual body. This lower spreading of
the flow causes a more rapid boundary layer growth and consequently reduced
heating. This effect is confirmed and amplified in the a = 53.5' case (Figure
11) where the predictions are about 50% low in the forward region In addition,
it appears that the history of this thicker boundary layer also causes an
underprediction of the heating along the aft regions.
The results of the third assumption, that of the small crossflow theory,
show excellent agreement with data in the forward regions for all three angles
of attack. There is a tendency to overpredict wing region heating at a = 15O.
At a = 30°, only the first half of the wing region shows any disagreement.
Finally, at a = 53.S0 the small crossflow prediction is low along the latter half of the body by an average of 25C. It is noted that Marvin, et. al.,
(Reference 113) suspect that the final data point at a = ~ 3 . 5 ~ (flagged in
Figure 11) is in a region of transition to turbulent flow. BLIMP-predicted
values of Reg at this location were of the order of 200 to 250. Based on the
correlations given by ~ a s e k ~ ~ for the dependence of the transition parameter
Reg/IMe. (Re/L)1°=2 with angle of attack, transition might be expected at Reg
of approximately 150 to 175. If the flow were laminar, the heat flux at this
last body location would be less than that of the previous upstream location.
For a blunt axisymmetric body assumption, accounting for an entropy layer (as opposed to an isentropic expansion) results in an increase in the predicted
heating rates of about 108, 15%.and 25% xespectively for the a = lSO, 30'
and 53.5' cases. Inclusion of an entropy layer appears to be only a small
impravement over the isentropic expansion case, but even these small effects
may have significant influences on a shuttle vehicle design. A measure of
the rate at which flow from the high entropy stagnation region shock is
swallowed by the boundary layer is obtained by comparing the shock wave angle
through which the boundary layer edge streamline originated. For a = 30%, this
shock wave angle is plotted in Figure 13 as a function of the surface coordinate s
and it can be observed that the swallowing process is essentially complete before
s % 0.1 ft. The continuing but slow decrease of the shock angle, and thus the entropy, is primarily due to the continual decrease of the shock angle along
the entire body length (see Appendix 2 ) . ~igures 14 and 15 show typical boundary
layer velocity profiles (solid lines) for a = lSO and 30' at x/L of 0.7 to 0.8.
The dashed curve represents the stock angle of the streamline as a function
of y , the normal distance fron the wall. These plots also indicate that the
entropy layer has been swallowed well forward on the body.
Predictions combining the entropy layer with the small-cross flow theory
were not made as this system.has not been incorporated into the current version
of the BLIMP code (this is conceptually straightforward but requires some code
modification). However, by extrapolating the available results, small-crossflow
with entropy layer will slightly increase the isentropic small-crossflow results
for c = lSO and 30° and should substantially improve the prediction over the
latter half of the body for the a = 53.5" case.
Turbulent Flow Results
The turbulent flow results for a = 30° are presented in Figure 11.
Transition to turbulent flow in the. wind tunnel experiments of Reference 113 was reported to occur at x/L = 0.54. Transikion to the fully turbulent model in
the BLIMP codc was made to occur at the ;me location, thus the apparent
instantaneous increase in turbulent heating levels. The interesting result
here is the effect of the e:ltropy leyer on turbulent heating. Whereas in the laminar region the increase, above the isentropic axisjrmmetric value, is limited
to about lo%, in the turbulent region the increase is 40% resulting in good agree-
n:ent with the data downstream of the actual trans4.tion region. This difference
in relative effects is most probably due to the greater dependence of transport
properties near the wall on edge conditions in turbulent flow as compared to
laminar flow. At the transition station edge gradients still persist primarily
due to shock curvature along the body +!s shown by Figure 16 which includes the
laminar velocity profile and the streamline shock :..~gle at the final laminar
station. The reported transition momentum thickness Reynolds number, Re , in Reference 113 was approximately 450. Predicted values for the fixed transition
location were 470 for the axisymmetric rur: and 330 for the small crossflow theory.
FIGURE 14 VELDCI-ry AND ~ T ~ Z E A M L ~ N E 0 3 1 6 1 N PROFILES, o( = 1 5 O (LAMINAR FLOW)
FI&uEE 15 V E L O C ~ T Y AND ~ ~ E A M L I N E OR1 GIN PIZOFILES, d = 30' (LAMINAP FLON)
OTHER RESULTS
Equations (6-2) and (6-3) were derived with a Newtonian pressure assump-
tion. For shuttle configurations, this is not quite correct so that the validity
of these equations might be questioned. In Equation (6-3), the Newtonian
assumption appears solely in the second term of the h2 coefficient where ~ ~ c o s ~ 0 ~
represents the Newtonian pressure ates on the plane of symmetry. At a given
BB the actual pressure will be slightly higher; in particular, as BB approaches 90'. pTcos2eB goes to zero whereas the real pressure does not. Consequently,
replacing the term ~ ~ c o s ~ 0 ~ with the actual surface pressure would increase
the magnitude of the corresponding term in Equation (6-3).
To investigate this point further, two solutions were obtained for the
a = 50' spreading factor case, one with an h2 distribution based on ~ ~ c o s ~ 0 ~ and
the other on the measured pressure on the symmetry axis. The two results were
identical from the stagnation point to an x/L of 0.1. From this point to the
beginning of the fli at x/L = 0.2 the heat fluxes based on the measured pressure
increased to 5 percen; above the ~ ~ c o s ~ 0 ~ value. Over the remaining (wing)
portion of the body the measured values were from 0 to 2 percent higher than
P ~ c o s * ~ ~ . Thus, it is concluded that for the shuttle configuration, this effect
is of a minor order. For all angles of attack, the reported results are
based on use of the local pressure which should overpredict local heating by
no more than 2 percent.
The effect of transverse curvature in the aft region was evaluated by
assuming that RT was (a) infinite and (b) equal to that of a cone locally
tangent to the surface and coaxial with the angle-of-attack axis. For a = 30°,
the tangent cone approximation resulted in an increase in heat fluxes from 5-10%.
Additional comparisons which would be beneficial are shown in Figures 17,
18 and 19. In these figures the BLIMP exisymmetric blunt body predictions are
compared with the tangent cone approximations recommended i.n Section 4.4. The
data of Reference 113 are also included. At all three angles-of-attack (15O,
30°, 53.5') the tangent cone approximations are slightly higher than the BLIMP
prediction but both are in generally good mutual agreement. However, at
a = 53.5' both predictions are noticeably lower than the data and would suggest
that other factors such as three dimensional entropy swallowing or variable
transverse curvature effects are important, especially at high angles-of-attack.
These effects are presently approximated in the BLIMP code. Although these
approximations do not necessarily represent optimum approaches, improved
methods will probably rely on the development of near-exact inviscid solutions
such as the work of References 21 and 22. Consequently an important consideration
in the development of a boundary layer code is the inclusion of an option for
integral coupling with an inviscid code.
RECOlSl3ENDATIONS FOR FURTHER DEVELOPMENT OF HEATING PREDICTION CAPABILITY FOR SHUTTLE VEHICLES
Although it is possible ?o predict the approximate magnitude of the heat
transfer to be expected on a delta shuttle vehicle, the reusability concept
dictates a higher level of accuracy. The following are some recommendations
on methods and procedures to improve accuracies of present methods.
1. The BLIMP code should be modified to account for nonequilibrium
homogeneous chemistry.
2, Nonequilibrim BLIHP should be used to evaluate the influence of
locally noncatalytic su-faces on downstream heat transfer.
3, The BLIMP code should be used to evaluate the influence of various
approximations of streamline spreading and entropy swallowing
effects for typical shuttle flight cocditions.
4. X c r i t i c ~ l e~ai~inaliori c1113 avaluaiioll of transition-to-turbulence
infomation should be performed to define a viable transition criteria,
The present BLIMP code has options for integral coupling with inviscid flow
field calculations and surface reaction chemistry. Any modifications and
improvements to this code should be developed so as to retain these options.
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Rubesin, M. W., "A M ~ d t f ~ e d Reynolds Analogy for the Compressible Turbulent Boundary Layer on a Flat Plate," NACA TN 2917, March 1953.
Thomas, A. C., et.,:l., "Application of Ground Test Data to Reentry Vehicle Design," AFFDL TI:-229, January 1967.
Savage, R. T. and Jaeck, C. L., "Investigation of Turbulent Heat Transfer at Hypersonic S?eeds, Vol. 1, Analytical Methods," AFFDL TR 67-144, Vol. 1, December 1967.
Colburn, A. P., "A Method of Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction," Trans. AIChE, Vol. XXIX, p.174- 211, 1933.
Reshotko, E. and Tucker, M., "Approxinate Calculation of the Compressible Turbulent Boundary Layer with Heat Transfer and Arbitrary Pressure Gradient," NACA TN 4154, December 1957.
Sommer, S. C. and Short, B. J., "Free Flight Measurements of Turbulent- Boundary-Layer Skin Friction in the Presence of Severe Aerodynamic Heating at Mach Numbers from 2.8 to 7.0," NACA TN 3391, March 1955.
Pearce, B. E., "A Comparison of Simple Turbulent Heating Estimates and Boundary Layer Transition Criteria with Application to Large, Lifting Entry Vehicles," NASA TM X-52876, Vol. 1, p. 485-508, July 1970.
Yopkins, E. 3. and Inouye, M., "An Evaluation of Theories for Predicting Turbulent Skin Friction and Heat Transfer on Flat Plates at Supersonic and Hypersonic Mach Numbers," A I M Jour., - 9, 6, p.993-1003, June 1971.
Beckwith, I. E. and Gallagher, 3. J., "Local Heat Transfer and Recovery Temperatures on a Yawed Cylinder at a Mach Number of 4.15 and High Reynolds Numbers," NASA TR R-104, 1961.
Thomas, A., Perbachs, A., and Nagel, A., "Advanced Reentry Systems Heat Transfer Manual for Hypersonic Flight," AFFDL-TR-65-195, Ocrober 1966.
Katzen, E. D., et.al., "Static Aerodynamics, Flow Fields and Aerodynamic Heating of Space Shuttle Orbiters," NASA TM X-52876, Vol. 1, p. 142-194, July 1470.
Young, C. H., Reda, D. C., and Roberge, A. M., "Transitional and Turbulent Heat Transfer Correlations for a Lifting Entry Vehicle at 24 .= 10," iiASii TM X 52876, Vol. 1, p. 418-444, July 1970.
- . -
Anon.jGenera1 Dynamic/Convair), "Space Shuttle, Volume 4: Technical Analy-. sis and Performance, Finai Technieai Report,'' NASA Ci3 102552, V o l . 4, Sec- tion 4, October 1969.
Guard, F. L. and Schultz, H. D., "Space Shuttle Aerodynamic Heating Consid- erations," ASME 70-aT/SpT-10, June 1970.
Creager, M. O., "The Effect of Leading Edge Sweep and Surface Inclination on Hypersonic Flow Field Over a Blunt Flat Plate, NASA Memo 12-26-58A, 1959.
Thomas, A. C., "Interference and Radiation Blockage Effects on Surface Tem- peratures of Composite Flight Vehicles," NASA TK X-52876, Vol. I, p.390- 417, July 1970.
Masek, R. V., "Boundary Layer Transition on Lifting Entry Vehicle Configura- tions at High Angles of Attack," NASA TM X-52876, Vol. 1, p. 445-462, July 1970.
Moote, 3. D., "A Minimum Heating Flight Mode for High Lateral Range Space Shutcle Entries Including the Effects of Transition," NASA TM X-52876, p.531-546, July 1970.
Marvin, J. G. and Sheaffer, Y. S., "A Method for Solving the Nonsimilar Laminar Boundary Layer Equations Including Foreign Gas Injection,'' NASA TN D-5516, November 1963.
Masek, R. V. and Forney, J. A . , "An Analysis of Predicted Space Shuttle Temperatures and Their Impact on Thernal Protection Systems," NASA TM X-2272, Vol. 1, p. 75-96, April 1971.
Schlichting, H., Boundary Layer Theory, Chap. XXI, McGraw Hill, New York, New York, 1960.
Maise, G., "Lee-Side Heating Investigations of Simple Body-Like Configura- tions," NASA TM X-2272, Vol. I, p. 289-309, April 1971.
Hefner, J. N. and Whitehead, A. H. Jr., "Experimental Lee-Side Heating Studies on a Delta-Wing Orbiter," NASA TM X-2272, Vol. I, p. 267-288,-~~ril 197 0.
Der, J. Jr., "A Study of General Three-Dimensional Boundary-Layer Problems by an Exact Numerical Method," AIAA Jour., - 9, 7, p.1294-1302, July 1971. Schlichting, H., 'A Survey of Some Recent Research Investigations on Bound- ary ~a~ers-and Heat ~ransfer," Jour. App. Mech., - 38, Ser. E, 2, p. 289-300, Jcne 1971.
Chapman, D. R., "A Theoretical Analysis of Heat Transfer Regions of Sepa- rated Flow," NACA TN 3792, 1956.
Schadt, G. H., "Aerodynamic Heaticg Problems and Their Influence on Earth Orbit Lifting Entry Spacecraft," AIAA Paper No. 68-1126, October 1968.
Feldman, S., "Hypersonic Gas Dynamic Charts for Equilibrium Air," AVCO RR-40, January 1957.
Lees, L., "Laminar Heat Transfer Over Blunt-Nosed Bodies at Hypersonic Flight Speeds," Jet Propulsion, p.259-274, April 1956.
Dunavant, J. C., MTnvesciyarion of Heat Transfer and 2ressures on Sighly Swept Flat and Dihedraled Delta Wings at Mach Numbers of 6.8 and 9.6 and Angles of Attack to 90°," NASA TM X-688, June 1962.
Bartlett, E. P., "An Evaluation of Design Analysis Techniques for High Performance Ballistic Vehicle Graphite Nose Tips, Appendix C, Boundary Layer Transport Phenomena," AFML-TR-69-73, Vol. 111, January 1970.
~liigge-~otz, I. and Blottner, F. G., "Computation of the Compressible Laminar Boundary-Layer Flow including Displacement Thickness Interaction using Fi~ite-Difference Methods," Stanford University, Div. of Eng. Mech., TR 131, January 1962.
Fannelop, T. K. and ~lGgge-~otz, I., "Two-Dimensional Viscous Hypersonic Flow over Simple B1ur.t Bodies including Second-Order Effects," Stanford University, Div. of Ens. tlech., TR 144, June 1964.
Davis, R. T. and Flugge-Lotz, I., "Laminar Compressible Flow Past Axisym- metric Blunt Bodies (Results of a Second-Order Theory)," Journal of Fluid Mechanics, - 20, 4, pp. 593-623, 1964.
Tong, H., "Multicomponent Nonequilibrium Boundary Layer Program," Boeing Rept. D2-23929-1, 1966.
Adams, J. C. , Lewis,. C. H., et. al., "Effects of Chemical Nonequilibrium, Mass Transfer, and Viscous Interaction on Spherically Blunted Cones at Hypersonic Conditions," AEDC-TR-69-237, January 1970.
84. Davis, R, T., "Numerical Solutions to the Viscous Shock-Layer Blunt Body Problem with Inert Gas Injection," Sandia SC-CR-70-6162, January 1971.
85. Lewis, C. H., Anderson, E. C., and Miner, E. W., "Nonreacting and Equilibrium CHemically Reacting Turbulent Eoundary-Layer Flows," AIAA Paper No. 71-597, June 1971.
86. Lew, H. G., "The Ionized Flow Field over Re-entry Bodies," General Electric TIS R67 SD 70, December 1967.
87. Braun, E. R., "Effects of a Fully Catalytic Wall on a Non-equilibrium Boundary Layer Including Ablation Products," General Electric TIS 70 SD 253, June 1970.
88. Dellinger, T. C., "Nonequilibrium Air Ionization in Fully Viscous Shock Layers," A I M Paper No. 70-806, June 1970.
89. Smith, A.M.O. and Clutte;:, D. W., "Machine Calculation of a Compressible Laminar Boundary Layers," AIAA Journal, - 3, 4, pp. 639-647, April 1965.
90. Smith, A.M.O. and Jaffe, N. A., "~eneral Method for Solving the Laminar Nonequilibrium Boundary-Layer Equations of a Dissociating Gas," AIAA Jcurnal, - 4, 4, pp. 611-620, April 1966.
91. Jaffe, N. A., Lind, R. C., and Smith, A.M.O., "Solution to the Binary Diffusion Laminar Boundary-Layer Equations with Second-Order Trans- verse Curvature," AIAA Journal, - 5, 9, pp. 1563-1569, September 1967.
92. Mayne, A. W., Jr., Gilley, G. E., and Lewis, C. H., "Binary Boundary Layers on Sharp Cones in Low Density Supersonic and Hypersonic F~ow,'' AIAA Paper No. 68-66, January 1968.
93. Smith, A.M.O., and Cebeci, T., "Numerical Solution of the Turbulent- Boundary-Layer Equations," Douglas Report No. DAC 33735, May 1967.
94. Cebeci, T., Smith, A.M.O., and Wang, L. C., "A Finite-Difference Method for Calculating Compressible Laminar and Turbulent Boundary Layers," McDonnell Douglas Rept. No. DAC-67131, March 1969.
95. Cebeci, T., "Calculation of Compressible Turbulent Boundary Layers with Heat and Mass Transfer," AIAA Jour., - 9, 6, pp. 1091-1097, June 1971.
* 96. Bushnell, D. M. and Beckwith, I. E., "Calculation of Nonequilibrium
Hypersonic Turbulent Boundary Layers and Comparisons with Experimental Data," AIAA Jour., - 8, 8, pp. 1462-1469, August 1970.
97. Herring, H. J. and Mellor, G. L., "A Method of Calculating Compressible Turbulent Boundary Layers," NASA CR-1144, September 1968.
98. Pallone, A. J., Moore, J. A. , and Erdos, 3. I., "Nonequilibrium, on similar Solutions of the Boundary-Layer Equations," AIAA Jour., - 2, 10, pp. 1706-1713, October.
99. Luceri, J. A,, "Reentry Environment and Systems Technology (REST) Semiannual Progress Report 1 January-30 June 1966, Vol. 11 -- Aerophysics Appendices," BSD-TR-66-231, Vol. 11, July 1966.
* Nonequilibrium as used here means turbulent flows with pressure gradient.
100. Erdos, J., and Pallone, A. J., "Interaction of a Chemically Reacting Laminar Boundary Layer and an Ablating Surface, Part I; Analysis," ARL 68-0029, February 1968.
101. Spalding, D. B., and Patankar,. S. V., Beat and Mass Transfer in Boundary Layers, C..R.C. Press: Cleveland, 1968.
102. Kays, W. M., "Heat Transfer to the Trmspired Turbulent Boundary Layer," paper to be presented at ASME Heat Transfer Conference 1971 (Copies available upon request from Stanford llniv., Mech. Eng. Dept.)
103. Mayne, A. W., Jr,, and Adams, J. C., Jr., "Streamline Swallowing by Laminar Boundary Layers in Hypersonic Flow," AEDC-TR-71-32, March 1971.
104. Kendall, R. K., and Bartlett, E. P., "Nonsimilar Solution of the Multi- component Laminar Boundary Layer by an Integral-Matrix Method," AIAA Jour., 6, 6, pp. 1089-1097, June 1968. -
105. Aerotherm Corporation, Mountain View, California, "User's Manual Boundary Layex Integral Matrix Procedure, Version C (BLIMPC)," Report KO. UM-70-20, June 1970.
106. Galowin, L. S., and Gould, H. E., "A Finite Difference Method Solution of Non-similar, Non-equilibrium Air, Laminar and Turbulent Boundary Layer Flows," Parts I, 11, and 11, General Applied Science Laboratories, Tech. Rept. No. 422, March 1964.
107. Galowin, L. S., and Gould, H, E., "Examples of Exact Numerical Solution of t h e Lssiiiar Eoundary Layci- Equations by Explicit Finite Difference Methods," General Applied Science Laboratories, Tech. Rept. No. 502, February 1965.
108. Mondrzyk, R. J., "Nonsimilar, Nonequilibriurn Laminar Boundary Layers for Sharp Conical Bodies," Boeing Rept. D2-36398-1, December 1965.
109. Lee, J. T., Langlet, T. J., and Moore, J. A., *'Nonequilibrium Laminar Boundary Layer Calculations on a Slightly Blunted 11.5 Degree Cone- Cylinder," TRW Systems Rept. 4509-6015-T, January 1966.
110. Moore, J. A., and Lee, J. T., tlDiscontinuous Injectian of Inert Gases into the Non-equilibrium Laminar Boundary Layer," TRW Systems Rept. 06388- 6018-R000, July 1967.
111. Chenoweth, D. R., "Finite Difference Solution of the Boundary-Layer Equations," Sandia SCL-DR-67-32, May 1967.
112. Marvin, J. G., NASA Ames Research Center, Moffett Field, California, private communication, Jcly 1971.
113. Marvin, J. G., et. al., "Flow Fields and Aerodynamic Heating of Space Shuttle Orbiters," NASA TM X-2272, pp. 21-73, April 1971.
114. North American Rockwell, "Wind Tunnel Model Contours-134 B , " NAR Drawing 3-930, 30 October 1970.
115. North American Rockwell, "Wind Tunnel Model Contours-129," NAR Drawing 9992-22, 1 June 1970
116. Hearne, L. F.,e~hin, J. H., and Woodruff, L. W., "Final Report: Study of Aerothermodynamic Phenomena Associated with Reentry of Manned Spacecraft," Lockheed Missile and Space Co., Sunnyvale, Calif., May 1966.
APPENDIX 1
LOCAL SURFACE INCIDENCE RELATIVE TO FREE STREAM DIRECTION
APPENDIX 1
L O W SURFACE INCIDENCE RELATIVE TO FREE STREAM DIRECTION
Local Sur face Incidence of Winy Surface
Defining p e r t i n e n t ang le as
a = v e h i c l e r e f e rence angle o f a t t a c k - a = ang le beixeen chord l i n e and re fe rence l i n e - . -. a = l o c a l s u r f a c e i n c l i n a t i o n with r e spec t t o chord l i n e
4 = l o c a l d i h e d r a l angle
then a t a = 0 , t h e u n i t v e c t o r which i s tangent t o t h e s u r f a c e and i r e s i n t h e
chord s e c t i o n plane i s
h h - h
u1 = c o s ( h + E ) i - s i n 1 j
and t h e u n i t vec to r which is tangent t o t h e su r f ace b u t which l ies i n t h e p lane
perpendicular t o the r e f e rence l i n e i s
A h A
Uz = s i n 4 j + cos 4 k
The cross product of these two vectors i s normal t o the loca l tangent plane and i n normalized form i s
where
%
Rotating U3 through an angle of attack a
A
The length of the projection of Uj onto the x-y plane is
and
cos t$ L = - N
The u n i t normal vec to r which d e f i n e s the l o c a l tangent p lane d t angle of a t t a c k
a is then
A - s i n ( a + ; + z ) c o s 4 - c o s ( a + & + z ) c o s 4 A cos (a + + I ) s i n t!
N U4 - - - - - - N j + N
A A
The d o t product of U4 with i i s t h e s i n e of t h e e f f e c t i v e rncidence. Defining y a s t h e e f f e c t i v e incidence
- - - cos 4 s i n ( a + G a ) s i n y - N
where t h e s i g n was changed t o y i e l d a p o s i t i v e angle.
For smal l + :his reduces to
s i n y = cos s i n ( u + G + g)
Local SurX?.ce Incidence of Outboard Fin Surface
Defining p e r t i n e n t ang le s a s
a = angle between f i n chord line an3 v e h i c l e symmetry plane
4 ' = angle of tilt of f i n - a' = l o c a l p r o f i l e angle r e l a t i v e tc chor2 l i n e
then a t a = 0 , t h e u n i t vec to r a t a n angle 4 ' from t h e v e r t i c a l and i n a p lane
perpendicular to t h e f i n c>ord l i n e i s
A A A A
U1 = - s i n @ ' s i n a i + cos 4 ' j + s i n O'coa a k
The u n i t vecLor which i s tangent t o the f i n su r f ace and l ies i n a h o r i z o n t a l p lane is
The cross product of U1 and U2, a f t e r normal izat ion y i e l d s t h e u n i t normal
vector which de f ines t h e tangent plane. Thus
A
Uj = - cos 4 ' s i n ( a + ) s i n $ ' s i n a s i n (a + a') N1
+ s i n $ ' cos a c o s ( a + a') ' c o s ( a + s t ) N1
where
~f = cos 0' + s i n 2 + ' s i n a s i n ( a + P ' ) + cos a cos(a + 2 ) I ' A
Using t h e same p r o c e d u r e a s before , Uj i s r o t a t e d through an angle a. The u n i t normal vec to r which d e f i n e s t h e l o c a l su r f ace tangent a t ang le of a t t a c k
a is then
A A A
u4 = - L c i n ( a + F)5 - L c o s ( a + 6 ) j A + cos 4' c o s ( a + a m ) N1
where L2 = COS'O's~~' (a + ow)
NZ + f {sin ( ' s i n a s i n ( a + a ' )
1 N1
+ s i n $'cos a c o s ( a + a') I tan 5 = cos S a s i n ( a + a ' )
s i n $ ' s in a s i n ( a + G') + s i n 4'cos a cos(a + z') A A
Again, t h e dot product of U4 and i i s the s i n e o f t h e e f f e c t i v e angle of incidence
yl. With the s i g n changed t o y i e l d a p o s i t i v e angle ,
and L and 5 can be expressed as
[cos ti8 t a n 4 ' J
For small angle $ t h i s set of equations reduce t o
s i n y t = L s i n ( 6 + a )
s i n 5 = d ~ ,
True Sweep Angle of King Leading Edge
Define
C = semi-apex angle of wing
$ = dihedra l angle
Then consider the u n i t vector which is the a x i s of a hypothet ica l c y l i n d r i c a l
leading edge. This u n i t vec to r is
A
cos 5 s i n 5 t a n $ A A i + - s i n 5 U1 = -- j + -
where
Rotating t h i s vec tor through an angle of a t t a c k a , the u n i t vector becomes
where
n A A s i n 5 A
U2 = L cos (a - A 2 ) i - L s i n ( a - A 2 ) j + k N1
t a n A, = t a n 5 t a n 9 - . . n
the d o t product of U2 and i is the cosine of t h e e f f e c t i v e angle of a t t a c k A . Thus
cos A = L cos (a - A2)
For small angles 9 this reduces t o
I cos A - cos C cos a
Trile Sweep Angle of Outboard Fins
Define
6 = nominal sweep angle of fins measured on vehicle symmetry plane
also +' and a are as previously defined. Then the unit vector which represents
the axis of a hypothetical cylindrical leading edge is
A sin 6 A cos 6 A - i + - j +
(sin 6 tan a + cos 6 tan 4 ' ) ; - N1 N1 N1
where
N* = 1 + (sin 6 tan a + cos 6 tan +'12
Rotating through an angle of attack a, the unit vector hec9mes
A 1 1 n
U2 = - A sin 6 tan a + cos 6 tan 4 ' sin(a + 6)i + - cos(a + 6)j + N1 N1 N1
and the effective angle of attack of the fin leading edge is
cos A ' = sin(a + 6) 1 + (sin 6 tan a + cos 6 tan @'Ii
APPENDIX 2
INPUT DATA FOR BLIMP TEST CASES
INP
UT
DA
TA
FOR BLIMP
TE
ST
CA
SES
a =
lS
O
Su
rfa
ce
Shock Wave
rs (f t)
0 . 0
0.0
00
99
5
0.0
01
96
0
.00
28
75
0
.00
37
6
0.0
04
5
0.0
05
83
0
.00
78
3
0.0
09
74
0
.01
39
0
.01
96
0
.02
79
0
.03
68
0
.05
16
3
.06
83
0
.11
4
0.1
69
0
.22
4
Data
h2 (it
)
0.0
0
.00
09
95
0
.00
19
7
0.0
02
91
0
.00
38
0
0.0
04
75
0
.00
56
0
0.0
06
65
0
.01
00
0
.01
40
0
.01
87
0
.02
70
0
.04
00
0
.05
50
0
.09
50
0
.13
2
0.3
10
0
.53
0
0.8
60
2
.20
4
.40
8
.70
1
3.0
1
7.0
2
1.3
2
6.3
S (
ft)
0.0
0
.00
1
0.0
02
0
.00
3
0.0
04
0
.00
5
0.0
06
0
.00
7
0.0
10
0
.01
3
Data
0, (
de
g)
90
.0
82
.2
74
.5
68
.2
62
.7
59
.1
53
.5
47
.2
42
.9
37
.6
34
.2
31
.0
20
.5
25
.5
23
.3
21
.1
19
.7
18
.7
P/P,
1.0
00
0
.97
25
0
.89
0
0.7
30
0
.65
5
0.6
00
0.
54'1
0
.50
2
0.4
20
0
.37
5
0.3
58
0
.33
5
0.3
12
0
.29
3
0.2
60
0
.23
2
0.1
88
,
0.1
56
0
.13
2
0.1
14
0
.10
4
0.0
90
0
.08
8
0.0
88
0
.08
8
0.0
88
rOk
(ft)
0.0
0
.00
09
95
0
.00
19
6
0.0
02
87
5
0.0
03
76
0
.00
45
0
0.0
05
17
0
.00
58
3
0.0
07
83
0
.00
97
4
0.0
16
1 0
.01
15
9
0.0
20
0
.02
5
0.0
30
0
.04
0
0.0
47
1
b.0
70
0
.09
0
0.1
11
0
.16
0
0.2
19
0
.32
5
0.4
31
0
.53
7
0.6
43
0
.74
9
0.0
13
9
0.0
16
7
0.0
19
6
0.0
25
0
0.0
27
9
0.0
36
8
0.0
44
1
0.0
51
6
0.0
68
3
0.0
86
5
0.1
14
0
.14
15
0
.16
9
0.1
97
0
.22
4
(r7b m b m m w m w m t-mmm a w m m d m m a m m m F O ~ O d N m w m r - O a m a m m O w m d 0 0 0 0 0 0 0 ? l r l m u 1 c o h l m w h l m
0 0 0 0 0 0 0 0 0 0 0 0 0 ~ N m w m . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
INP
UT
D
AT
A
FO
R
BL
IMP
T
ES
T
CA
SES
a =
53.5O
Surface
*
S (ft)
0.0
0.0005
0.001
0.002
0.003
Data
h2 (f t)
0.0
0.0005
0.001
0.00235
0.0047
0.0093
0.030
0.073
0.175
0.900
4.2
450.0
5900.0
3.2
x lo5
6.1
x lo6
3.3
x lo7
1.6
x 10'
2.9
x 10'
4.2
x 10'
5.4
x 10'
6.7
x 10'
8.0
x 10'
9.1
x 10'
1.0
x 10'
Shock Wave
r, (ft)
0.0
0.00833
0.01667
0.02500
0.03333
0.04167
0.05000
0.06667
0.08333
0.1250
0.1667
0.2500
0.4167
0.5417
0.6250
0.7208
0.8333
0.9167
1.0000
rOk
(ft)
0.0
0.000475
0.00095
0.00190
0.00285
P/P,
1.0 .9955
.987s1
.978-
.970
,964
.950
.940
.932
i915
.900
.848
.810
.788
,761
.741
.72
4
.716
.717
.719
-730
.739
.540
.409
Data
8, (deg) I
90.0
83.6
78.3
74.0
70.5
67.6
65.2
62.2
60.4
60.0
59.4
59.0
58.4
58.1
58.5
59.6
56.7
55.5
55.3
0.004
0.00380
0.006
1 0.00570
0.008
0.010
0.015
0.0211
0.050
0.080
0.1065
0.160
0.2095
0.3155
0.4215
0.5275
0.6335
0.7395
0.8455
0.9518
1.0508
0.00760
0.00950
0.0140
0.0200
0.0475
0.0740
0.0980
0.144
0.186
0.2705
0.355
0.4395
0.524
0.6085
0.693
0.775
0.845
APPENDIX 3
SPREADING FACTOR FOR NONZQUAL BODY CURVATURE
APPENDIX 3
SPREADING FACTOR rOR NONEQUAL BODY CURVATURE
The variable Rc in Equation (6-2) is not ambiguous since it applies o n l y
to a sphere, the application being the Apollo reentry heat shield. For shuttle
application, Equation (6-2) is not valid since it does not account for the un-
equal surface curvatures which exist forward of the wing region. The deriva-
tion of Reference 116 assumes Newtonian flow along a streamline and transverse
to it in arriving at an expression for the crossflow pressure gradient in the
meridianal direction.
where 2 is the distance from the plane of symetry.
For a non-spherical surface, it is necessary to return to the basic
assumption of Newtonian flow, namely
where $ is the angle between the local surface normal and the free
stream velocity vector.
Referring to the sketch, the circle with radius RT represents
t h e tangent circle i n t h e plane perpendicular t o t he windward s t reaml ine
(gene ra to r ) , z = 0. A r o t a t i o n i n t h i s p lane by an angle $, which i s equiva-
l e n t t o some displacement z from t h e plane of symmetry, r e s u l t s i n a change
i n t he angle $. This angle may be found by viewing t h e tangent c i r c l e both
i n planform and on edge. The p o i n t Q r e p r e s e n t s t h e t i p of t h e r a d i u s
vec tor a t some angle 6. The angle t h i s vec to r makes with t h e plane perpendicu-
lar t o t h e v e l o c i t y v e c b ~ r i s simply
and thus
a = s in - ' d
F
bu t from t h e above sketches w e have the r e l a t i o n
and
thus
JI ( B ) = 90° - sin-' [ c o s ~ cos$(0) I
Returning t o the Newtonian pressure expression, Equation (A-2 ) , it i s now
poss ib l e t o f i n d the pressure gradient i n the d i rec t ion normal tC the symmetry
streamline.
or
where
but cos (90' - y) = s iny
and
theref ore
. s i n (s in- 'x) = x
1;~w f o r small z and thus 8 , quat ti on (14) becomes i n the l i m i t
w i t h
thus
z B 9 ~ and m = z
i n the l i m i t as z goes to zero.
?f the 10,:al symmetry plane pressure in Equation (A-8a) is the Newtonian
value then
T S C --Ll- = P(0.0) = PT c0s2$ ( ~ ~ 0 )
and Equation (102) becomes
where $ ( 0 ) and Bg are equivalent quantities. , , -8 \
ThusI Equation (6-2) is replaced for non-spherical shapes by
(A-I 9 )
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